(In)eﬀiciency in Private Value Bargaining with Naive Players:
Theory and Experiment∗
Alex Possajennikov†

Rene Saran

University of Nottingham

University of Cincinnati

June 2023

Abstract
The paper investigates two-player double-auction bargaining with private values ina setting with discrete two-point overlapping distributions of traders’ valuations. We
characterize parameter settings in which there exists a fully eﬀicient equilibrium, andshow that if there are traders that behave naively, i.e., set bid or ask equal to theirvaluation, then there is no equilibrium achieving full eﬀiciency. We conduct an experiment to test the theoretical possibility that the presence of naive traders can reduce
eﬀiciency. We find, however, that eﬀiciency is not lower in the presence of naive traders.
Subjects mostly set bid/ask prices strategically but they do not coordinate on a fully
eﬀicient equilibrium and the extent of strategic behavior is not different in the presenceof naive players. We show that a learning model of noisy strategy adjustment explains
the observed behavior better than other (equilibrium or non-equilibrium) models.
Keywords: bargaining with private values, double auction, eﬀiciency
JEL Codes: C72, C78, C91, D82∗

We thank the British Academy for financial support and Centre for Decision Research and Experimental
Economics (CeDEx) at the University of Nottingham for access to laboratory facilities. The paper benefited
from audience comments at various seminar and conference presentations. We also thank the editor andanonymous referees for their comments.
†
Corresponding author: School of Economics, University of Nottingham, University Park, Nottingham
NG7 2RD, United Kingdom. Tel: +44 115 9515461. Email: alex.possajennikov@nottingham.ac.uk. Web:https://sites.google.com/view/alex-possajennikov.

11 Introduction
Many goods, e.g., cars, homes, and novelty items in a bazaar are traded in bargaining
encounters involving one buyer and one seller. The buyer and seller are often asymmetricallyinformed: each trader knows his/her own reservation price (value or cost) but not that of
the other trader. In such private value bargaining negotiations, a trader faces the basic
tradeoff between the probability and terms of trade: misrepresenting one’s reservation pricecan improve the terms of trade but can reduce the probability of trade. Strategic traders
optimally respond to this tradeoff, which can result in an ineﬀicient outcome, i.e., thegood is not traded even though the buyer’s value exceeds the seller’s cost (Myerson andSatterthwaite, 1983).
Experimental evidence, however, suggests that some people behave naively, i.e., they
reveal their private information even in the presence of monetary incentives to misrepresent
it.1 Saran (2011) incorporates the possibility of naive traders in private value bargaining
to show that a mechanism designer who tailors the mechanism to the proportion of naive
traders can improve eﬀiciency. However, if the mechanism is fixed, then the impact of naive
traders on eﬀiciency in bilateral trading is ambiguous. The first-order effect of introducing
naive traders (i.e., the change in outcome while keeping the behavior of strategic tradersconstant) is to improve eﬀiciency (because they always trade when it is eﬀicient). But thesecond-order effect (i.e., the change in outcome when strategic traders react optimally tothe presence of naive traders), could be such that it either enhances or reduces eﬀiciency.2
In this paper, we further investigate possible effects of naive traders on eﬀiciency andindividual behavior in private value bargaining. To that end, we consider the double-auction
mechanism and a simple setting with discrete (two points) overlapping but non-identical
distributions of values and costs. In the double auction, the buyer submits a bid while theseller submits an ask and trade happens if and only if the bid weakly exceeds the ask, at theprice midpoint between the bid and the ask. In this setting, we take as naive the behavior
1See, for example, Fischbacher and Föllmi-Heusi (2013), Gneezy et al. (2018), and Abeler et al. (2019)
for evidence of this in simple one-person reporting decisions, and Gneezy (2005), Lundquist et al. (2009) andSerra-Garcia et al. (2013) for interactive situations. These studies suggest aversion to lying (and aversion to
being seen lying) as an explanation of naive behavior. Bounded rationality, i.e., inability to understand theimplications of revealing one’s private information could also explain naive behavior in some circumstances.
For example, although people say that they are concerned about privacy, they willingly reveal personal
information on the internet (e.g., Spiekermann et al., 2001).
2
For example, Saran (2012) shows that the presence of naive traders in a double auction with pre-playcommunication can improve eﬀiciency since the strategic traders will act less strategically in the pre-playcommunication stage lest they lose the chance to trade with the naive traders in the double-auction stage ata favorable price. Without pre-play communication, Saran (2011) shows that the presence of naive traders
in a double auction can reduce eﬀiciency when the intervals of values and costs overlap at only one point.

2

of setting the bid equal to the private value, or the ask equal to the private cost, directly
revealing the private information.
We are specially interested in the theoretical possibility that the presence of naive traders
can reduce eﬀiciency in this environment, which will be the case if the second-order effectis negative and dominates the trivially positive first-order effect. To have a chance ofobserving such a negative second-order effect, we characterize parameter settings in which
fully eﬀicient equilibria exist without naive traders. However, as the proportion of naive
traders increases, strategic traders have an incentive to increase their strategic behavior
and the eﬀicient equilibria disappear. Thus, if the traders coordinate on the most eﬀicientequilibrium, then, in theory, the introduction of naive traders should reduce eﬀiciency.
Our confidence in observing this theoretical possibility is buttressed by previous experimental studies on the double auction which show that the outcomes are consistent withthe most eﬀicient equilibrium. For the case of identical continuous uniform distributionsof values and costs, the linear equilibrium suggested by Chatterjee and Samuelson (1983)
is the most eﬀicient (even though not all trades with positive surplus are achieved in it).3
Although there are many other equilibria of the double auction in that case (Leininger etal., 1989; Satterthwaite and Williams, 1989), experimental data typically conform to thelinear equilibrium (Radner and Schotter, 1989; Valley et al., 2002; McGinn et al., 2003;
Ellingsen et al., 2009).
Even in our simple discrete setting, the double-auction mechanism has multiple equilibria, both in the absence and presence of naive traders. In the absence of naive traders, theeﬀiciency attained in equilibrium can range from zero to full eﬀiciency. Similarly, the equilibrium eﬀiciency has a wide range in the presence of naive traders, although it is strictly
positive on the lower end and strictly less than full eﬀiciency on the upper end. Dependingon which equilibria are played, eﬀiciency could be higher or lower as the proportion of naive
traders increases. Hence, equilibrium theory per se does not provide a clear prediction regarding the effect of naive traders on eﬀiciency in our setting. We conduct an experiment
to bring more clarity on this issue.
In our experiment, we consider two treatments. In one (control) treatment, each subject
always plays against another (randomly rematched) subject in the experiment, with one
subject assigned the role of the buyer while the other the role of the seller. In the othertreatment, with some probability, a subject, instead of being matched with another subject,
is matched with a computer that will set ask/bid equal to cost/value. Subjects know thispossibility but at the time of making their decisions, they do not know whether they will
3

Myerson and Satterthwaite (1983) show that the linear equilibrium is in fact constrained eﬀicient for thecase of identical continuous uniform distributions, i.e., there is no other bargaining mechanism which would
lead in equilibrium to a higher ex-ante expected gains from trade.

3

be matched with a computer or with another subject. In this way, the chance of playing anaive opponent increases with the introduction of the artificial naive subjects, and we cananalyze the effect of this on the behavior of human participants.
In the standard double auction, traders set prices, but an equivalent game can also beframed as a direct mechanism.4 Theoretically this does not affect the set of equilibria, but
the direct mechanism may be more conducive to naive play if subjects are averse to lying
since it is clear what telling the truth means there. Since such a framing can affect theshare of traders playing naively, we also compare the performance of the double auction
framed as a bid-ask setting and as a direct mechanism with the same allocation rule.
The experiment results are mixed, with no decrease in eﬀiciency in the presence of
naive traders. Eﬀiciency in fact increases for some settings but this is solely due to thepositive first-order effect of artificial naive traders rather than due to a change in strategic
behavior by human subjects. The subjects do not appear to coordinate on the fully eﬀicientequilibrium in the setting without artificial naive players, and there is no clear effect of
the change in the proportion of naive traders on behavior, either measured by the extent
to which ask/bid differs from cost/value or by the proportion of naive plays by humanparticipants: thus, the results point toward a zero second-order effect. The direct mechanism
frame has an effect to make some traders behave naively more often (thus providing someevidence of aversion to direct lying), but this effect is not suﬀicient to change eﬀiciency.
We also look at the patterns of subjects’ individual decisions and try to identify what
determines the extent of strategic shading or exaggeration. Most subjects do behave strategically, but the extent of strategic behavior does not change significantly across different
settings to influence the aggregate measures of eﬀiciency. We find that subjects often follow
what worked for them in the past, but they also respond to incentives to some extent asbids and asks adjust towards the best response to the observed history.
Since we minimize repeated-game effects in our experiment by randomly rematchingsubjects in each period, the fact that the observed history significantly affects individual
behavior suggests that a learning model might be appropriate to explain behavior. We
specifically consider the experience-weighted attraction (EWA) learning model of noisy best
response (Camerer and Ho, 1999). The model is initialized with behavior based on low
levels of strategic sophistication in the level-k model (e.g., Stahl and Wilson, 1994).5 Thismodel (which also takes into account possible disutility of deviation from private cost/value)
indeed has a better fit to the data than static (equilibrium or non-equilibrium) models
4

In the direct mechanism, players make direct statements about their values/costs. Trade happens if thebuyer’s statement exceeds the seller’s, at the price equal to the midpoint between the stated value and cost.
5
More precisely, the initial attractions of the EWA model are set equal to the expected payoff against
the uniform distribution of strategies of the other player (level-1).

4

with a comparable number of parameters, or an alternative model with preferences forfairness (see Section 6.2). Simulations of the EWA model reproduce many features of theexperiment results, also finding that eﬀiciency and the extent of strategic behavior do notchange much with a change in the proportion of naive players.6 Our analysis suggests that
non-equilibrium models like level-k and learning dynamics may be more useful predictorsof play in the double auction than equilibrium and its eﬀiciency properties.
The rest of the paper is organized as follows. In Section 2, we present the theoreticalmodel and conditions that characterize the fully eﬀicient equilibrium. In Section 3, we
discuss the design of our experiment and provide our hypotheses. We present the results
related to eﬀiciency and average strategic behavior in Section 4. In Section 5, we showhow history of play influences individual behavior. In Section 6, we present the dynamic
model of learning and compare its fit with some other models. We conclude in Section 7
and collect proofs in the Appendix.

2 Double Auction Bargaining with Discrete Private Values
2.1

Model

There are two risk-neutral traders, a single seller (s) of an indivisible object facing a single
buyer (b). The seller’s cost c of producing the object can be either low c of high c̄ with
qs = 12 being the probability of c. Similarly, the buyer’s value v for the object can be either
low v or high v̄ with qb = 12 being the probability of v̄. The seller privately knows her cost
while the buyer privately knows her value. We assume c < v < c̄ < v̄.7
The traders use the double auction as the trading mechanism. In this mechanism, thetwo traders simultaneously submit a bid (buyer) and an ask (seller). If the buyer’s bid zb
weakly exceeds the seller’s ask zs , then the object is traded at price p = 12 (zb + zs ). Thenthe buyer’s payoff is v − p, while the seller’s payoff is p − c. On the other hand, if zb < zs ,then there is no trade and each trader obtains a zero payoff.
Each trader i can have one of two dispositions ti : strategic (str) or naive (n). In the
double auction, the naive buyer sets bid equal to her value while the naive seller sets ask
equal to her cost. The strategic trader can set any bid/ask in the double auction. The
traders do not know each other’s disposition. The probability that a trader is naive isε ∈ [0, 1), which is independent of the trader’s value/cost.
6

That the model also fits initial choices is consistent with level-k model being a good predictor of initial
responses in games (see Crawford et al., 2013 for a survey). Crawford (2021) and Kneeland (2022) apply
the level-k theory to private value bargaining from the point of view of mechanism design.
7
Chatterjee and Samuelson (1987) consider such a setting for a model of alternating offer bargaining; Feriand Gantner (2011) run an experiment based on their model.

5

2.2

Equilibrium and Eﬀiciency

Let Zsts : {c, c̄} → R and Zbtb : {v, v̄} → R denote the pure strategies of, respectively, theseller and buyer in the double auction. For the naive traders, by assumption Zsn (c) = c and
Zbn (v) = v. Thus we only need to analyze the strategies of strategic traders Zcstr and Zbstr .
Let q ts ,tb (·, ·) : {c, c̄} × {v, v̄} → [0, 1] denote the probability of trade between the tradersdepending on their dispositions and values. Given that c < v < c̄ < v̄, trade is eﬀicient if it
happens in all cases except when cost is c̄ and value is v. Therefore trade is fully eﬀicient
if q ts ,tb (c̄, v) = 0, q ts ,tb (c, v) = q ts ,tb (c, v̄) = q ts ,tb (c̄, v̄) = 1 for all ts and tb .
For any ε ∈ [0, 1), define11(2(1 − ε)c + (1 + ε)v) ;B(z2 , c, v) =(z2 + 2c − εv) ;
3−ε
3−ε
1
C(z2 , c, v, v̄) =(εv̄ + 2(2 − ε)c − εv − (1 − ε)z2 ) ;3−ε
11D(c, v) =(2(1 − ε)v + εc) ;E(c̄, v̄) =(2(1 − ε)c̄ + εv̄) ;
2−ε
2−ε
1
1
F (c̄, v̄) =(2(1 − ε)v̄ + (1 + ε)c̄) ;G(z1 , c̄, v̄) =(z1 + 2v̄ − εc̄) ;
3−ε
3−ε
1
H(z1 , c̄, v̄, c) =(εc + 2(2 − ε)v̄ − εc̄ − (1 − ε)z1 ) .
3−εA(c, v) =The following proposition characterizes fully eﬀicient equilibria of the double-auction game.
Proposition 1. Suppose ε ∈ [0, 1). Strategies (Zsstr , Zbstr ) are a fully eﬀicient equilibrium
if and only if Zsstr (c) = Zbstr (v) = z1 and Zsstr (c̄) = Zbstr (v̄) = z2 , wheremax{A(c, v), B(z2 , c, v), C(z2 , c, v, v̄)} ≤ z1 ≤ D(c, v)
andE(c̄, v̄) ≤ z2 ≤ min{F (c̄, v̄), G(z1 , c̄, v̄), H(z1 , c̄, v̄, c)}.
The proof is in the Appendix. In a fully eﬀicient equilibrium, the low-cost strategic
seller and the low-value strategic buyer (and correspondingly the high-cost strategic seller
and the high-value strategic buyer) agree on one price z1 (and correspondingly z2 ). Then
there is also trade if the low-cost strategic seller meets the high-value strategic buyer andno trade when the low-value strategic buyer meets the high-cost strategic seller, and tradeis also eﬀicient if a strategic trader meets a naive opponent (between naive traders, tradeis always eﬀicient). The inequalities in the proposition ensure that the strategic traders do
not find it profitable to deviate from z1 and z2 . Depending on the values c, v, c̄, v̄ and ε, thefully eﬀicient equilibrium may or may not exist.
If ε = 0, then the inequalities in Proposition 1 reduce to 32 c + 13 z2 ≤ z1 ≤ v andc̄ ≤ z2 ≤ 23 v̄ + 13 z1 . With ε > 0, there are more deviations to consider than with ε = 0,
6since a strategic trader may attempt to extract full surplus from her naive counterpart.
Therefore, the conditions for the existence of fully eﬀicient equilibria are more stringent for
ε > 0 than for ε = 0. Even though a fully eﬀicient equilibrium may exist without naive
traders, it may cease to exist if a positive proportion of naive traders is present.
Irrespective of whether a fully eﬀicient equilibrium exists or not, there are multiple
equilibria in the double-auction game, both in pure and mixed strategies. At the lower
end of eﬀiciency, there is a “no-trade” equilibrium, in which strategic traders do not trade
between themselves. For example, both types of strategic buyer bidding c and both typesof strategic seller asking v̄ is an equilibrium for any ε if 2v̄ + c ≤ 3c̄ and 3v ≤ v̄ + 2c, as wehave in our experiment settings. If ε = 0, then such an equilibrium implies zero eﬀiciency
but if ε > 0 eﬀiciency is positive and increasing with ε since naive traders trade between
themselves.

2.3

Settings in the Experiment

We use the results from Proposition 1 to choose the parameter settings in the experiment.
In the main treatment, we consider c = 10, c̄ = 70, v = 30, v̄ = 90 (coded as “20” by
the difference v − c = v̄ − c̄ = 20). Without naive traders, there is a unique fully eﬀicientequilibrium in this setting. If naive traders are present in any proportion, the fully eﬀicientequilibrium disappears.
Proposition 2. Suppose that c = 10, c̄ = 70, v = 30, v̄ = 90.
i. If ε = 0, there exists a unique fully eﬀicient equilibrium Zsstr (10) = Zbstr (30) = 30 and
Zsstr (70) = Zbstr (90) = 70.
ii. If ε ∈ (0, 1), there is no fully eﬀicient equilibrium.
The proof is in the Appendix. It shows that for ε = 0, there is only one set of values
z1 and z2 that satisfy the conditions for the existence of a fully eﬀicient equilibrium. In
this equilibrium, the low-value buyer and the high-cost seller obtain zero surplus, since they
trade at price equal to their value or cost. The equilibrium is thus a knife-edge case in thesense that the low-value buyer and high-cost seller have only weak incentives to agree tosuch trades.
This knife-edge property explains why full eﬀiciency is not attainable in equilibrium inthe presence of naive traders. Instead of agreeing to bid her value, the strategic low-value
buyer can increase her surplus by shading her bid and trading with the naive low-cost seller.
Similarly, the strategic high-cost seller would like to exaggerate her ask to get more fromthe naive high-value buyer. The presence of naive traders thus makes these strategic traders7behave more strategically (increase shading/exaggeration). As a result, there are no values
for z1 and z2 that ensure eﬀicient trade. Therefore, if the players manage to coordinate on
the fully eﬀicient equilibrium in the absence of naive traders, then the introduction of naive
traders in the experiment should reduce eﬀiciency.
The knife-edge property of the fully eﬀicient equilibrium when ε = 0 can make it diﬀicult
to coordinate on it because some types of traders earn zero surplus in this equilibrium. Also,in the experiment (some) players might expect (some) others to play naively, either because
those others are not expected to understand strategic incentives of the game, or they are
expected to care about honesty or eﬀiciency. Thus, even before the artificial introduction ofnaive traders in the experiment, some players’ underlying beliefs might be that a positive
proportion of opponents are naive.8
We therefore consider another setting, c = 5, c̄ = 70, v = 30, v̄ = 95 (coded as “25” forv − c = v̄ − c̄ = 25). When ε = 0, this alternative setting has several fully eﬀicient equilibria
that result in positive surplus for all types of traders. At the same time, some of these fully
eﬀicient equilibria remain for small proportions of naive traders – thus making it possible
to obtain full eﬀiciency even when players’ underlying beliefs are that a positive proportionof opponents are naive. But it is still the case that for suﬀiciently large presence of naive
traders there is no fully eﬀicient equilibrium, and thus the artificial introduction of naive
traders, as we do in the experiment, may reduce eﬀiciency even in this alternative setting.
Proposition 3. Suppose that c = 5, c̄ = 70, v = 30, v̄ = 95.
i. If ε = 0, there exist several fully eﬀicient equilibria.
√
ii. If ε ∈ (0, 1), a fully eﬀicient equilibrium exists for ε ∈ (0, ε̄], where ε̄ = (11− 101)/5 ≈0.190, and does not exist for ε ∈ (ε̄, 1).
The proof is in the Appendix. Fully eﬀicient equilibria in this case exist for all valuesof ε up to approximately 0.19. When ε exceeds 0.19, the strategic traders’ incentives toextract surplus from the naive traders becomes overwhelming, and it is not possible toachieve eﬀicient trade between the strategic traders anymore.9
All the above propositions are based on self-interested risk-neutral strategic traders.
However, many plausible alternative formulations of preferences preserve a fully eﬀicient8

Nevertheless, even in this case one can expect that the increase in ε leads to lower eﬀiciency. In Section
S1.1 of Supplementary Materials online, we show that in a class of simple symmetric mixed equilibria (which
exist for reasonable values of ε, viz. ε < 0.367), maximum eﬀiciency decreases monotonically with ε.
9
In Section S1.2 of Supplementary Materials online we show that maximum eﬀiciency in this setting
equals full eﬀiciency for ε ≤ ε̄ and then decreases monotonically with ε in the class of simple symmetric
mixed equilibria that exist for ε < 0.411.

8equilibrium whenever it exists. For example, if players are risk-averse, a fully eﬀicientequilibrium remains, since any deviation from it either reduces a player’s ex-post payoff orincreases the variance of the expected payoff. Similarly, if players are eﬀiciency concerned,
the fully eﬀicient equilibrium remains. If players have a cost of lying (or from setting bid/ask
different from their value/cost), a fully eﬀicient equilibrium also remains.10
Since there are multiple equilibria in the double auction (including those asymmetricbetween the seller and buyer roles), what equilibrium is played, if any, may depend on manyfactors. Also, players may not necessarily coordinate on an equilibrium, act strategically or
have selfish preferences. The theory thus does not offer unambiguous prediction. However, it
seems intuitive that, with more naive players, strategic players find it profitable to act morestrategically (i.e., increase shading/exaggeration) as the presence of naive traders dilutes
the underlying tradeoff between the probability and terms of trade: for any given belief inthe behavior of the strategic opponent, a marginal increase in shading/exaggeration has asmaller negative effect on the probability of trade and a greater positive effect on the termsof trade.11 Our choice of parameter settings makes it possible that such second-order effects
can lead to fewer trades in equilibrium, and this is what we try to see in the experiment.

3 Experiment Design
Players in the double auction make decisions knowing the distribution of possible costs andvalues, and knowing their own cost (if seller) or value (if buyer). We consider experimental
treatments differing along the following dimensions: the absence or presence of naive traders,different parameter settings discussed above, and the framing of interactions. Our main
treatment difference is in the artificially induced proportion of naive traders. In treatments
labelled “S” (for “strategic”) all subjects are matched in pairs and play the double auction.
10

All these alternative preferences can create additional fully eﬀicient equilibria, also for the setting withnaive traders. However, as we will see below, fully eﬀicient equilibria are not what is observed in theexperiment.
11
For instance, suppose the high-value strategic buyer believes that the strategic seller’s strategy is given
by some distribution of asks Fs , which has the density fs . Then, in the absence of naive seller, the strategic∫zbuyer’s expected payoff when she bids zb > c̄ is 0 b (v̄ − (zb + zs )/2)fs (zs )dzs . Hence, the effect of a marginalincrease in shading (i.e., marginal reduction in zb ) equals Fs (zb )/2 − (v̄ − zb )fs (zb ). The first term, Fs (zb )/2,is the improvement in the terms of trade due to a reduction in the price of any trade. The second term,−(v̄ − zb )fs (zb ), is a negative effect (representing the loss at the margin) due to a decrease in the probabilityof trading. In contrast, in the presence of naive seller, the high-value strategic buyer’s expected payoff when()∫z
she bids zb > c̄ is (1 − ε) 0 b (v̄ − (zb + zs )/2)fs (zs )dzs + ε (1/2)(v̄ − (zb + c̄)/2) + (1/2)(v̄ − (zb + c)/2) .
()(
Hence, now the effect of a marginal increase in shading equals (1 − ε)Fs (zb ) + ε /2 − 1 − ε)(v̄ − zb )fs (zb ).
Thus, the improvement in the terms of trade is greater while the negative impact on the probability of tradeis smaller in the presence of naive seller. This argument generalizes to other cases too.

9

In treatments “N” (for “naive”), subjects are told that with probability 0.25 their decision
will be matched with that of a computerized opponent who sets bid equal to the buyer’svalue or ask equal to the seller’s cost.12
The distributions of costs and values follow the two settings discussed in the previous
section. One setting (“20”) involves c = 10, v = 30, c̄ = 70, v̄ = 90. Recall that in thissetting there exists a fully eﬀicient equilibrium if the proportion ε of naive traders is 0,
but not for any other value of ε < 1. Increasing ε as is done in the N treatment may
thus decrease eﬀiciency compared to the S treatment. However, as mentioned earlier, wemight fail to observe the fully eﬀicient equilibrium in this setting for two reasons: firstly,
because some types of traders earn zero surplus, making it potentially diﬀicult to coordinate
on this equilibrium; secondly, it is possible that (some) subjects’ underlying beliefs about
opponent’s naiveté are positive. We therefore also consider setting “25” with c = 5, v = 30,c̄ = 70, v̄ = 95. In this setting, for values of ε less than 0.19, there exist fully eﬀicientequilibria in which all types of traders earn positive surplus. Thus, the S treatment in thissetting offers the opportunity to attain full eﬀiciency with positive surplus to all types oftraders as long as the subjects’ underlying beliefs regarding their opponent’s naiveté arenot too high. But full eﬀiciency becomes unattainable in equilibrium once the proportion ofnaive traders is artificially increased in the N treatment.13 Hence, if we assume coordination
on a fully eﬀicient equilibrium in treatment S, then we can expect eﬀiciency to be lower intreatment N than S.14 We thus have our first main hypothesis:
Hypothesis 1. Eﬀiciency in treatment S is higher than in treatment N.
Compared with a fully eﬀicient equilibrium in treatment S, the presence of naive tradersin treatment N in theory makes strategic sellers with high cost and strategic buyers withlow value (traders with the non-extreme values) increase shading of bids or exaggeration
12

To be precise, subjects are still matched in pairs in this treatment. After they have chosen their bid/ask,with probability 0.25, both subjects in a pair are rematched with computerized opponents who play naively.
Thus, each subject faces a 0.75 probability of being matched with the human opponent and 0.25 probabilityof being matched with a naive computerized opponent. For each subject, the probability of meeting a naive
opponent in the N treatment increases. This correlated matching avoids replacing subjects’ bids with “naive”
bids, but makes it impossible for two naive (computerized) traders to be matched, unlike in the theoreticalmodel. To capture the effect of such matches and make eﬀiciency measures comparable with the theoreticalones, we adjust measures of expected eﬀiciency in the experiment as explained in Section 4.1.
13
For reasonable beliefs about ε being close to 0.1 (based, for example, on Valley et al., 2002, who find thatabout 10% of subjects make ask/bids equal to their cost/value in their no-communication treatment), the N
treatment implies the increase in the probability of meeting a naive trader by 0.75 · 0.1 + 0.25 − 0.1 = 0.225.
14
Even if beliefs about naive traders are not necessarily ε = 0 in treatment S, eﬀiciency still can be
expected to decrease monotonically with ε, as we demonstrate in Section S1 of Supplementary Materials
online, using simple symmetric mixed equilibria.

10

of costs, since they can get a higher expected payoff by exploiting naive traders.15 Again,
assuming coordination on a fully eﬀicient equilibrium in treatment S, we have our second
main hypothesis:
Hypothesis 2. There is a higher extent of strategic behavior (shading/exaggeration) by thetraders with the non-extreme values in treatment N than in treatment S.
The effect of introduction of naive traders on the equilibrium behavior of strategic lowcost sellers and high-value buyers (traders with the extreme values) is less clear cut. For
example, in the unique fully eﬀicient equilibrium in setting “20” with ε = 0, the high-value
buyer and the high-cost seller coordinate on the price of 70 while the low-value buyer andthe low-cost seller coordinate on the price of 30. With ε = 0.25, there exists an equilibrium
in which the high-value buyer and the low-cost seller coordinate on the price of 28, thelow-value buyer bids 10 and the high-cost seller asks 90. Compared to the fully eﬀicientequilibrium, the low-cost seller now acts less strategically whereas all other types of traders
act more strategically.16 Due to this ambiguous effect, we do not state a formal hypothesis
comparing the extent of strategic behavior by the traders with extreme values.
Our main hypotheses presume coordination on a fully eﬀicient equilibrium in treatment S. In general though, there are multiple equilibria in the double auction and, without
further assumptions/refinements, equilibrium theory does not offer an unambiguous prediction, as discussed in the last section. Beyond equilibrium, at a more basic level, an increasein the probability of naive traders has a positive first-order effect on expected eﬀiciency
for the trivial reason that now a greater proportion of traders – the naive ones – are notshading/exaggerating their bids/asks. If the strategic traders react “instinctively” to theincrease in naive behavior (i.e., while assuming that the behavior of the strategic opponentis held constant), then they have a greater incentive to shade/exaggerate because, as mentioned in the last section, a greater proportion of naive traders dilutes the tradeoff between
probability and terms of trade. The resulting second-order effect could increase strategic
behavior (from both non-extreme and extreme value traders), which could cause eﬀiciency
to decrease in matches where both traders are strategic. Thus, to the extent that players areprone to reacting instinctively, we can expect (at least some) human traders to behave morestrategically with a higher proportion of naive traders, possibly leading to lower eﬀiciency in
15

Even if beliefs about naive traders are not necessarily ε = 0 in treatment S, we can expect strategic
behavior (shading/exaggeration) of traders with the non-extreme values to increase monotonically with ε, as
we demonstrate in Section S1 of Supplementary Materials online, using simple symmetric mixed equilibria.
16
There is also an equilibrium in which the high-value buyer and the low-cost seller coordinate on theprice of 72, the low-value buyer bids 10 and the high-cost seller asks 90. Here, compared to the fully
eﬀicient equilibrium, the high-value buyer acts less strategically whereas all other types of traders act morestrategically.

11human matches in treatment N, which is consistent with our hypotheses. Statistically, thehypotheses can be tested and average treatment effects identified by comparing eﬀiciency
and strategic behavior in the two treatments (provided that the assignment of subjects totreatments is random).
We also vary the framing of the interaction. In the double-auction framing (coded“BA”), the seller names an ask price and the buyer names a bid price. In addition to thisstandard framing, we consider the direct mechanism framing (coded “DM”). In this framing,
subjects are asked to report their cost or value. The payoffs are calculated according to the
double-auction rule: trade occurs if the reported value is above the reported cost, at theprice in the middle between the reported value and cost. The difference between BA and
DM frames is only in the wording of the instructions.17 Framing the interaction as a direct
mechanism makes it clearer what revealing one’s cost or value is, and therefore may affecthow subjects feel about exaggerating their cost or shading their value. Our expectation isthat experiment participants in frame DM play naively more often than in frame BA. If
other participants realize this, it can have a knock-on effect on their behavior, making themact more strategically in frame DM than in frame BA, and thus eﬀiciency may be higher inframe BA than in frame DM.
Each experimental session follows one of the settings. For example, a session coded as“BA-S-20” means a session using the double-auction frame, no artificial naive players, andparameters c = 10, v = 30, c̄ = 70, v̄ = 90. In each session, 16-18 subjects are divided into
two matching groups of 8-10 players. Within each matching group, in each period half ofthe subjects have one role (e.g. Seller) and the other half the other role. Table 1 gives thenumber of sessions, matching groups, and subjects for each treatment. In total, there are
16 sessions (32 matching groups) with 272 subjects, who all passed the control questions onunderstanding the instructions.
A session lasts 40 periods. In each period, a subject in one role is randomly matchedwith a subject in the other role. During a session, each subject keeps the same role for10 periods, then switches to the other role.18 After the matching, each subject’s value
or cost is drawn randomly and independently. The subjects then make their decisions by
inputting a number between 0 and 100 with at most two digits after decimal point19 andtheir payoffs are calculated.20 At the end of each period, subjects are told what the outcome
17

The full set of instructions is available in the Experiment Instructions file online.
Thus a subject could be e.g. Seller in periods 1-10, Buyer in periods 11-20, Seller again in periods 21-30
and Buyer again in periods 31-40.
19
Before making a decision, subjects could use a “Payoff Calculator” to calculate their own expected
payoffs for different possible asks/bids set by themselves and their opponents.
20
In the N treatment, it is first determined whether the subjects in a pair are rematched against computerized opponents and then payoffs are calculated.
18

12

Code

Sessions

Matching
groupsSubjects

Code

Sessions

Matching
groupsSubjects

BA-S-20
BA-N-20
BA-S-25
BA-N-25

3
3
2
2

6
6
4
4

52
52
34
32

DM-S-20
DM-N-20
DM-S-25
DM-N-25

2
2
1
1

4
4
2
2

34
32
18
18

Note: BA - bid/ask frame; DM - direct mechanism frame; S - treatment with ε = 0, N - treatment with ε = 0.25;
20 - setting “20”, 25 - setting “25”.

Table 1: Description of the experimental sessions
in their match is (but neither the opponent’s cost or value nor whether the opponent is acomputer).21 Payoffs are measured in points.
Experimental sessions were conducted in the Centre for Decision Research and Experimental Economics (CeDEx) laboratory at the University of Nottingham in March 2012.
The experiment was programmed in z-Tree (Fischbacher, 2007) and subjects were recruitedfrom the CeDEx database of experimental participants using the ORSEE software (Greiner,2015). They were students of various disciplines across the university.22 Sessions were assigned to treatments randomly, making the assignment of subjects to treatment random andallowing identification of average treatment effects. Together with reading the instructions,answering control questions, and filling in the post-experiment questionnaire with demographic information, each 40-period session lasted 90-120 minutes. Subjects were paid theiraccumulated earnings, converted from points to pounds at the rate £0.20 for 10 points insetting “20” and £0.15 for 10 points in setting “25” (to equalize payoffs, since the available
surplus is larger in setting “25”). The average payment per subject was £14.12 ($22.25 at
the time of the experiment), including a £5 show-up fee.

4

Experiment Results

4.1

Eﬀiciency

Before we present our results, we define how we measure eﬀiciency in our treatments. The
realized eﬀiciency would be the proportion of total available surplus that is captured by thetraders in our game. However, in treatment N this measure of eﬀiciency is affected by the21

Recall that our games have multiple equilibria. Having multiple rounds with feedback provides anopportunity to coordinate on a particular equilibrium, while random matching makes the game look closer
to a one-shot game.
22
Overall, 54% of the experiment participants were female; the average age was 20.0 years.

13

correlated matching with computerized opponents, which makes both subjects in a matchplay against the computerized traders.
To have eﬀiciency comparable with the theoretical model, in which the dispositions(being naive or strategic) of the players are independent rather than correlated, we calculate
expected probability of trade and expected eﬀiciency as if the dispositions were independently
determined. In each matched pair of human subjects, given the realized cost c and value
v and the subjects’ bid b and ask a, the expected probability of trade (for given ε) is
P Te = (1 − ε)2 · Ib≥a + (1 − ε)ε · Ib≥c + ε(1 − ε) · Iv≥a + ε2 · Iv≥c , where IX is the indicatorvariable for condition X (IX = 1 if condition X is satisfied and IX = 0 if condition X is notsatisfied). The expected surplus for this match is P Te · (v − c). These calculations allow tocapture the direct positive effect of naive traders on eﬀiciency, which is not available withthe correlated matching.23 The available surplus in a pair is (v − c) · Iv≥c . The overall
expected eﬀiciency is the ratio of the total expected surplus to the total available surplus
over all matched pairs.
As another measure of eﬀiciency we use the proportion of total available surplus thatwould have been captured by the traders if hypothetically all subjects were matched between themselves rather than with (possibly) a computerized opponent. Thus, the humanprobability of trade in a match with given bid b and ask a is P Th = Ib≥a . The surplus thatwould have been realized is P Th · (v − c). The human eﬀiciency is the ratio of this surplus
summed up over all matched pairs to the total available surplus over the pairs.
In treatment S (ε = 0), the expected eﬀiciency measure and the human eﬀiciency measure coincide (and they are also equal to the realized eﬀiciency). The theoretical analysis of
the situations played out in the experiment shows that if the strategic traders coordinate ona fully eﬀicient equilibrium in the absence of naive traders, then eﬀiciency (both expected
and human) can be higher without naive traders (treatment S) than with a sizable proportion of naive traders (treatment N). Also, if there are more naive plays in frame DM thanin frame BA, then it is possible that eﬀiciency is higher in the BA frame than in the DM
frame.
Figure 1 presents the comparison of eﬀiciency in treatments S and N. It shows the timeseries in the experiment of the three eﬀiciency measures – realized eﬀiciency in treatment
S labeled “S” (blue, lighter line), expected eﬀiciency in treatment N labeled “N-e” (red,23

For example, if the buyer’s value is 30 and the seller’s cost is 10, and the subject in the buyer’s role bids
20 while the subject in the seller’s role asks 75, then the expected probability of trade with ε = 0.25 andindependent dispositions is (0.75)2 (0) + (0.75)(0.25)(1) + (0.25)(0.75)(0) + (0.25)2 (1) = 0.25. The expectedsurplus is 0.25 · 20 = 5. In the experiment, because of the correlated matching, with probability 0.75 realized
surplus is 0, and with probability 0.25, realized surplus is 20 for the buyer and 0 for the seller. Averaging
over the pair, the realized surplus is 0.25((20 + 0)/2) = 2.5, smaller than the expected surplus.

14Efficiency in setting "20"Efficiency in setting "25"

1

1

0.95

0.95

0.9

0.85
0.8

0.9
0.85
0.8

0.75

0.75

0.7

0.7

0.65

0.65

0.6

0.6

0.55

0.55

0.5

0.5
1-5

6-10

11-15

16-20
S

21-25
N-e

26-30

31-35

36-40

1-5

N-h

6-10

11-15

16-20
S

21-25
N-e

26-30

31-35

36-40

N-h

Note: Blocks of periods on the horizontal axis; fraction of surplus realized on the vertical axis, aggregated acrossBA and DM frames. N-e: expected eﬀiciency in treatment N; N-h: human eﬀiciency in treatment N, as defined intext (in treatment S, expected and human eﬀiciency are the same).

Figure 1: Eﬀiciency measures in treatments S and N
darker line), and human eﬀiciency in treatment N labeled “N-h” (dotted line) – averaged
over blocks of five periods. The eﬀiciency measures are separated across the “20” and“25” settings since we expect eﬀiciency to be different in setting “25”, where positive trade
surpluses are larger. They are combined for the BA and DM frames, since the eﬀiciency
measures are very similar between the frames24 , and the regression results below do notfind significant differences between the frames.
As can be seen in the figure, all measures of eﬀiciency are clearly lower than full eﬀiciency
and do not change much over time. The expected eﬀiciency in treatment N appears higher
than the eﬀiciency in treatment S, especially in setting “20”. There appears to be little
difference between the treatments when comparing the eﬀiciency in treatment S with thehuman eﬀiciency in treatment N.
The eﬀiciency measures presented in Figure 1 are aggregate measures, combining situations with a high available surplus (v̄ − c) and a smaller surplus (v − c and v̄ − c̄) as wellas situations with negative trade surplus (v − c̄). Table 2 shows the expected and human
average probabilities of trades for the different surplus situations (combined across BA and
DM frames).25
From Table 2, the expected probability of trade is higher in treatment N than theprobability of trade in treatment S for all trade surpluses (even for the negative one v −c̄, although such trades happen only in less than 3% of all cases of negative surplus).26
24

The measures for each matching group in each treatment are presented in Section S2.1 of Supplementary
Materials online.
25
In the S treatment (ε = 0), the probabilities coincide (and equal to the realized proportion of trades),thus only one number is presented for treatment S.
26
It is also the case that the probability of trade with v̄ − c̄ is larger than with v − c. The explanation for15

Tradesurplusv̄ − cv̄ − c̄v−cv − c̄ (< 0)Setting “20”SN
P Te = P Th
P Te (P Th )
0.853
0.551
0.436
0.015

0.922 (0.861)
0.676 (0.528)
0.613 (0.438)
0.018 (0.023)

Setting “25”S
N
P Te = P Th
P Te (P Th )
0.947
0.622
0.479
0.008

0.947 (0.906)
0.650 (0.493)
0.646 (0.479)
0.043 (0.050)

Note: Averages from all rounds and across BA and DM frames. Expected and humanprobabilities of trade P Te and P Th are as defined in text.

Table 2: Probabilities of trade for different trade surpluses
However, the human probability of trade in treatment N is very similar to the probabilityof trade in treatment S for all gains of trade.
To test formally the effect of the treatments on the probability of trade, and thus oneﬀiciency, we regress the probabilities of trade on the treatment variables in Table 3. In thefirst two columns of the table, the dependent variable is the expected probability of trade(P Te ); in the last two columns, it is the probability of trade if all subjects were matchedbetween themselves (P Th ). The variable P Te can take values 0, 1, and also in between(see footnote 23), while P Th is a binary variable. Since we are interested in treatment
average effects rather than predicted probability, we use ordinary least square regressions
for both.27 Explanatory variables are dummy variables denoting treatment (Treatment N:
1 if treatment is N and 0 if S), frame (Frame DM: 1 if frame is DM, 0 if BA), or parametersetting (Setting “25”: 1 if setting is “25”, 0 if “20”). In a fuller model, we also include the
Period variable and dummy variables for available surplus (v̄ − c, v̄ − c̄, v − c̄; surplus v − cacts as the base category)28 .
The regressions show that the probability of trade, and thus eﬀiciency, is not affected byframe (BA or DM), and hardly by setting (“20” or “25”). The probabilities of trade increaseslightly over time but this effect is very small. Most significantly, the expected probability
this is discussed in the next subsection.
27
Average treatment effects (i.e., changes in the predicted value of the dependent variable as a treatment
variable changes from 0 to 1, averaged over all sample values of the other independent variables) from theprobit regression of P Th , reported in Section S2.2 of Supplementary Materials online, are similar to thosefound in the linear regression.
28
The regressions thus include observations with negative trade surplus v − c̄. If such observations are
excluded, the significance of the coeﬀicients remains the same and their magnitudes are similar, as reported
in Section S2.2 of Supplementary Materials online. Regressions on subsamples of situations with different
positive surpluses, also reported in Section S2.2 of Supplementary Materials online, show that the effects ofthe treatment variables are similar in situations of different surpluses.

16

Variables
Treatment N
Frame DM
Setting “25”Dependent variables
Expected trade probability P Te Human trade probability P Th
0.088∗∗∗

0.081∗∗∗−0.004−0.012(0.020)(0.021)(0.024)(0.025)−0.012−0.008−0.009−0.005(0.022)(0.023)(0.027)(0.029)0.041∗

0.031

0.044

0.035

(0.022)(0.023)(0.026)(0.028)Periodv̄ − cv̄ − c̄v − c̄
Constant
R20.001∗∗

0.001∗∗∗(< 0.001)(< 0.001)

0.371∗∗∗

0.430∗∗∗(0.020)(0.018)0.084∗∗∗

0.094∗∗∗(0.024)(0.026)−0.517∗∗∗−0.430∗∗∗(0.025)(0.023)0.464∗∗∗

0.465∗∗∗

0.462∗∗∗

0.420∗∗∗(0.023)(0.030)(0.026)(0.030)

0.01

0.48

< 0.010.38

Note: Treatment S, frame BA and setting “20” are the base categories in all columns; trade surplus v − c isthe base category in columns 2 and 4. Standard errors clustered by matching groups in parentheses, 5440
observations in 32 clusters. ∗∗∗ - significant at 1% level; ∗∗ - significant at 5% level; ∗ - significant at 10%
level.

Table 3: Effects of treatment variations on the probability of tradeof trade, and thus the expected eﬀiciency, is higher in treatment N than in treatment S, but
the probability of trade in matches between subjects is the same between the treatments.29
The introduction of naive traders thus raises the expected probability of trade but does notchange the probability of trade in matches between human subjects.
Result 1. (Eﬀiciency)• Expected eﬀiciency is higher in treatment N than in treatment S.
• Human eﬀiciency is not significantly different between treatments N and S.
• There are no significant differences in eﬀiciency between BA and DM frames.
29

Non-parametric tests at the matching group level, reported in Section S2.1 of Supplementary Materials
online also confirm this, especially for setting “20”.

17Thus our hypotheses related to eﬀiciency are not confirmed. Recall that if subjects were
to coordinate on the fully eﬀicient equilibrium in treatment S, then eﬀiciency in treatment
S will be higher than both the expected and human eﬀiciencies in treatment N. The results,
however, go in the opposite direction: the expected eﬀiciency in treatment N is higher thanthe eﬀiciency in treatment S in our experiment. Since we find that the human eﬀiciency
in treatment N is not different from the eﬀiciency in treatment S, the greater value of the
expected eﬀiciency in treatment N compared to the eﬀiciency in treatment S is mostly due tothe trivial first-order effect, i.e., the additional trades (that would have happened if traders’dispositions were determined independently) with and between computerized naive tradersrather than between human subjects.
An obvious reason for the failure to get higher eﬀiciency in the S treatment compared to
the N treatment is that subjects are not playing a fully eﬀicient equilibrium in treatment S.
The absence of a significant difference between eﬀiciency in the S treatment and eﬀiciency
in matches of human subjects in the N treatment, and between the probabilities of trade intreatment S and in matches of human subjects in treatment N, points also to the possibilityof a zero second-order effect, that is, subjects do not behave more strategically in treatment
N compared with treatment S. We analyze strategic behavior in more detail in the next
subsection.

4.2

Strategic Behavior

The previous subsection looked at the eﬀiciency measures in the experimental sessions,which are derived from players’ actual bids (b) and asks (a) (or statements about theirvalues (v) or costs (c) in frame DM). In this subsection we analyze players’ behavior inmore detail. In particular, we are interested by how much their bids and asks differ fromtheir value or cost (the extent of strategic shading/exaggeration). We term this difference(v − b for buyers and a − c for sellers) strategic behavior.
Figure 2 shows the time series of the average amount of shading and exaggeration in theexperiment. The data are presented separately for extreme- (top panels) and non-extremevalue traders (bottom panels), and they are also separated for the “20” (left panels) and“25” (right panels) settings, since in setting “25” the scope for ask/bids further away fromcost/value is larger. Behavior is further disaggregated between S (labeled ‘S-’) and N(labeled ‘N-’) treatments (while aggregated across BA and DM frames)30 .
Unsurprisingly, traders with the extreme values shade or exaggerate much more (on30

As regressions below show, there is no significant differences in strategic behavior between frames.
The averages of strategic behavior in all matching groups are presented in Section S3.1 of Supplementary
Materials online.

18

Strategic behavior of extreme-value traders in "20"Strategic behavior of extreme-value traders in "25"

30

30

28

28

26

26

24

24

22

22

20

20

18

18
1-5

6-10

11-15

16-20

21-25

S

N

26-30

31-35

36-40

1-5

Strategic behavior of non-extreme-value traders in "20"

6-10

11-15

16-20

21-25

S

N

26-30

31-35

36-40

Strategic behavior of non-extreme-value traders in "25"

12

12

10

10

8

8

6

6

4

4

2

2

0

0
1-5

6-10

11-15

16-20

21-25

S

N

26-30

31-35

36-40

1-5

6-10

11-15

16-20

21-25

S

N

26-30

31-35

36-40

Note: Blocks of periods on the horizontal axis; average strategic behavior (v − b and a − c) on the vertical axis,
aggregated across BA and DM frames. Top panels: extreme values (c and v̄); bottom panels: non-extreme values(v and c̄).

Figure 2: Extent of strategic behavior in treatments S and N
average, 25.97) than traders with the non-extreme values (4.35 on average). From the figure,there appears to be little difference in the average extent of strategic behavior between
treatments S and N. The extent of strategic behavior also stays roughly at the same levelacross periods. Thus our expectation of more strategic behavior in treatment N than intreatment S is not fulfilled. The figure also confirms that the play is not that close to thefully eﬀicient equilibrium: in “20” setting, non-extreme-value traders on average make bids
or asks closer to 25 for buyer with value 30 and 75 for seller with cost 70 (strategic behavior
5) and extreme-value traders shade value/exaggerate cost by more than 20.
To check the informal observations from the figure more formally, we use regression
analysis. In the regressions, the dependent variable is strategic behavior as defined above(a − c or v − b). Among the explanatory variables, Treatment N, Frame DM, and Setting“25”, as before, are the dummy variables describing the treatment, frame, and setting. We
run one set of regressions, separately for the subsamples of observations with the extreme
and with the non-extreme values, with only these variables. In another set we also include
the dummy variable Seller (equal to 1 if a subject is a seller in period t and 0 if the subject

19

Variable
Treatment N
Frame DM
Setting “25”Dependent variable: Strategic behavior
ExtremeNon-extreme−0.391−0.373

0.383

0.382

(1.812)(1.815)(0.713)(0.714)

1.192

1.115

−1.361∗−1.363∗(2.069)(2.081)(0.801)(0.801)

1.164

1.119

0.192

0.191

(2.086)(2.094)(0.768)(0.768)SellerPeriod5.668∗∗∗−0.064(1.182)(0.435)−0.0060.006(0.037)

Constant
Observations
R2(0.015)25.28∗∗∗

22.55∗∗∗

4.606∗∗∗

4.523∗∗∗(1.758)(1.758)(0.700)(0.700)5411< 0.015469
0.02< 0.01< 0.01

Note: Strategic behavior is a − c or v − b. Extreme: values c and v̄; nonextreme: values v and c̄. Treatment S, frame BA, and setting “20” arethe base categories in all columns; Buyer is the base category is columns
2 and 4. Standard errors clustered by 32 matching groups in parentheses.
∗∗∗ - significant at 1% level; ∗ - significant at 10% level.

Table 4: Effects of treatment variations on strategic behavior
if a buyer) and the variable Period. The results of the regressions are presented in Table 4.
The regressions confirm that the average strategic behavior is similar across treatments,
frames, and settings: none of the dummy variables for those are significant.31 Sellers appear
to be more strategic than buyers when having extreme values, while somewhat less strategic
with non-extreme values.32 No significant time trend is detected in the average strategic31

Non-parametric tests at the matching group level, reported in Section S3.1 of Supplementary Materials
online, find the same result. Regressions combining observations across subsamples, reported in Section
S3.3 of Supplementary Materials online, also produce the same results. Standard errors in the reported
regressions are clustered by matching group, since a subject’s behavior can depend on what happens in thewhole matching group, not only on the subject’s own characteristics and behavior. However, as reported in
Section S3.3 of Supplementary Materials online, the significance of the variables is the same with standard
errors clustered at the individual level.
32This difference is one explanation for the different probabilities of trade noted earlier: high-value buyers
and high-cost sellers trade more often because they are less strategic than low-cost sellers and low-value
buyers. While this difference between roles is somewhat surprising (the game is symmetric and all subjects
played an equal number of times in each role), Radner and Schotter (1989) also find a difference in buyer
and seller behavior in their experiment on symmetric double auction.

20

behavior.
Recall that we consider behavior naive if bid is set equal to value or ask equal to cost (or,in frame DM, actual value or cost is reported). This corresponds to strategic behavior being
equal to 0 and is captured by variable Naive, equal 1 if subject i played naively in periodt and 0 otherwise. Even though the average strategic behavior in the experiment is clearly
different from 0, subjects sometimes play naively: 7% of choices of traders with extreme
values and 28% of choices with non-extreme values are “naive”.33 Table 5 reports the resultsof linear regressions (again run separately for subsamples of extreme and non-extreme value
traders) for dependent variable Naive.34
The regressions find that there is significantly more naive play in frame DM (36%) thanin frame BA (24%) when traders have non-extreme values (v or c̄).35 With extreme values,
sellers play naively less often than buyers, while with non-extreme values the opposite istrue. There is a significant increase over time in naive play by traders with non-extreme
values (from 25% in the first 10 periods to 32% in the last 10).
Summarizing the findings about the average strategic behavior so far:Result 2. (Average strategic behavior)• There are no significant differences in the extent of strategic behavior across treatments
and frames; traders with non-extreme values play naively quite often, and more often
in frame DM than in frame BA.
• With extreme values, sellers play more strategically than buyers but with non-extreme
values, there is no difference in strategic behavior between sellers and buyers.
33

The large difference between these proportions shows that few subjects consistently played naively forboth extreme and non-extreme values. Since playing naively with an extreme value has very little strategic
sense, 7% can be taken as an indicator of naturally occurring naive players. This proportion is close to the
10% proportion of “naive” (ask/bid equal cost/value) plays in Valley et al. (2002) and can further justify
that reasonable beliefs of subjects about ε (not far from 10%) allow for a priori hypothesis about eﬀiciency
comparison between the S and N treatments. The proportions of naive play in each matching groups arereported in Section S3.2 of Supplementary Materials online.
34
Again, since our interest is in the treatment effects, not in prediction, linear regression coeﬀicients
approximate well the average treatment effects. Probit regressions, reported in Section S3.3 of Supplementary
Materials online, lead to similar results. So do regressions on the combined sample and with standard
errors clustered by individuals rather than matching groups, also reported in Section S3.3 of Supplementary
Materials online.
35
Non-parametric tests in Section S3.2 of Supplementary Materials online confirm the significance of thisdifference but do not find the differences between treatments S and N significant for either type of traders.

21Variable
Treatment N
Frame DM
Setting “25”Dependent variable: Naive behavior (Naive)
ExtremeNon-extreme
0.030

0.029

−0.014−0.015(0.019)(0.019)(0.041)(0.040)

0.008

0.009

0.119∗∗∗

0.120∗∗∗(0.020)(0.020)(0.041)(0.041)

0.002

0.002

0.004

0.005

(0.020)(0.020)(0.037)(0.037)

SellerPeriod−0.059∗∗∗0.080∗∗∗(0.011)(0.022)< 0.001

0.002∗∗(< 0.001)

Constant

0.072∗∗∗

0.243∗∗∗

0.159∗∗∗(0.013)(0.014)(0.048)(0.046)

Observations
R2(0.001)

0.051∗∗∗

5411< 0.01

5469
0.02

0.02

0.03

Note: Naive is a dummy variable equal to 1 if strategic behavior is 0, and 0
otherwise. Extreme: values c and v̄; non-extreme: values v and c̄. Treatment
S, frame BA, and setting “20” are the base categories in all columns; Buyer isthe base category is columns 2 and 4. Standard errors clustered by 32 matchinggroups in parentheses. ∗∗∗ - significant at 1% level; ∗∗ - significant at 5% level.

Table 5: Effects of treatment variations on naive behavior

5

Influence of History on Individual Behavior

From the previous section, the average strategic behavior is not affected much by the treatment, frame, or setting variations. Moreover, the average behavior tends to not vary muchover time. In this section, we focus on the time series of (disaggregated) individual behavior
to argue that individual strategic behavior is different across subjects and appears to be
influenced by history of play.
Figure 3 shows how the current strategic behavior in individual observations (on thevertical axis) relates to the previous strategic behavior (Previous SB) by the same individual
the last time he/she was in the same role and with the same cost/value (on the horizontal
axis; depending on the assignment of roles and the realization of cost and values, it may be
one or several periods ago). The left panel is for observations with extreme values and theright panel is for non-extreme values.
The figure shows that often the current behavior is the same as the previous one, butthere is also variability around the 45◦ line, as subjects change their behavior. We formally22

Strategic behavior of extreme-value traders

Strategic behavior of non-extreme value traders

100

30

10

Current Strategic Behavior

Current Strategic Behavior

80

60

-70

40

20

0
-20

-30

-10

10

30

-10

-30

-50

0
-20

-50

20

40

60

80

100

-70

Previous Strategic Behavior

Previous Strategic Behavior

Note: Strategic behavior is a − c or v − b. Current strategic behavior on the vertical axis; previous strategic behavior
by the same subject (in the same role and with the same cost/value) on the horizontal axis. All data for which
previous strategic behavior is available. Size of the circles corresponds to the number of observations. Left panel:extreme values (c and v̄); right panels: non-extreme values (v and c̄).

Figure 3: Current and previous individual strategic behavior
check the association of the current behavior with the experienced history by regressing
current strategic behavior on previous strategic behavior, while also including variables
measuring the effect of best response. Variable History-Best-Response is equal to how muchstrategic behavior should be if subject i were playing best response (BR) to the experienced
history up to period t.36 Similarly, variable Last-Best-Response is equal to the strategic
behavior at best response to the last observation of the opponent’s play by subject i inperiod t. These two variables measure the “optimal” strategic behavior in view of (longor short memory of) the player’s history. The three left columns of Table 6 report the
regression results, separately for samples with extreme and non-extreme values. The threeright columns report regressions of the variable Naive on previous naive behavior (equal 1
if previous strategic behavior is 0, and 0 otherwise), and variables measuring incentives todeviate from naive play: variable Payoff-Gain-History-BR is the payoff difference between
36

The variable is constructed as follows. In the double auction game the (seller’s, for example) best
response to observed bids is equal to either one of these bids or to the seller’s own cost. For example,
suppose that a seller with cost 10 observed bids 30, 70, 80, 30 in the previous periods. We calculate seller’s
payoff from asking 10, 30, 70, 80 against these bids; in this particular case, both 30 and 70 are best responses
to this experienced history. Such ties are broken in favor of the lowest strategic behavior; the value ofHistory-Best-Response is thus 30 − 10 = 20.

23

Dependent variable: Strategic behavior
Variable

ExtremeNon-extreme

Previous SB

0.660∗∗∗

0.406∗∗∗(0.017)(0.035)

History-BestResponse
Last-BestResponse

0.037∗∗∗

0.063∗∗∗(0.013)(0.017)0.027∗∗

0.079∗∗∗(0.010)(0.017)

Constant

6.096∗∗∗

1.531∗∗∗(0.748)(0.355)Observations

4867

4925

R2

0.44

0.19

Dependent variable: Naive
Variable

ExtremeNon-extreme

Previous Naive

0.588∗∗∗

0.607∗∗∗(0.045)(0.026)−0.0004−0.003(0.001)(0.005)−0.0003∗−0.005∗∗∗(0.0002)(0.001)0.040∗∗∗

0.132∗∗∗(0.007)(0.016)

Observations

4867

4925

R2

0.34

0.37

Payoff-GainHistory-BR
Payoff-GainLast-BR
Constant

Note: Strategic behavior is a − c and v − b. Naive is a dummy variable equal 1 if strategic behavior is 0, and0 otherwise. Extreme: values c and v̄; non-extreme: values v and c̄. Previous SB is strategic behavior by thesame subject last time the subject was in the same role and had the same cost/value. History-Best-Responseis the strategic behavior at the best response to experienced history; Payoff-Gain-History-BR is the differencein payoff between this best response and naive play. Last-Best-Response is the strategic behavior at the best
response to the last observation and Payoff-Gain-Last-BR is the difference in payoff between this best response
and naive play. Standard errors clustered by 32 matching groups in parentheses. ∗∗∗ - significant at 1% level;∗∗ - significant at 5% level; ∗ - significant at 10% level.

Table 6: Effects of experienced history on behavior
playing best response to the whole experienced history and playing naively,37 and PayoffGain-Last-BR is the payoff difference between playing best response to the last observationand playing naively.
All the variables representing a subject’s history in the game have significant coeﬀicientsin the regression reported in the three left columns. The extent of strategic behavior iscorrelated across time: the more strategic a subject was before, the greater the extent of
the current strategic behavior by the subject. The coeﬀicient on Previous SB is significantly
smaller than 1 though; thus players do not always keep playing the same strategy. An
increase in the distance between the best response to either the entire history of play or to
the last observation and the player’s value/cost increases the extent of strategic behavior
too. Subjects who played (more) strategically tend to continue playing (more) strategically,
but they also react to the history by changing towards best responses to their experience.
From the three right columns, playing naively is also correlated with playing naively in thepast. A higher payoff difference between playing best response to past experience (whole
history or last observation) and playing naively is associated with less naive play, although37This payoff difference is easy to calculate once we have determined the best response to any experienced
history, as discussed in the previous footnote.

24the effect is significant only for the payoff difference in the last observation.
We note the above findings about individual behavior in the next result:
Result 3. (Individual strategic behavior)• The extent of strategic behavior in the past is positively correlated with the extent
of strategic behavior now; strategic behavior also changes towards best responses toprevious observations (both distant in the past and immediate).
• Playing naively in the past is positively correlated with playing naively now.
Another take from the previous section is the relatively large proportion of observationsof naive play (also noticeable in Figure 3). In the double auction, naive behavior is weakly
dominated from purely selfish preferences point of view: reducing the bid (or raising theask) by the smallest available unit (0.01 in the experiment) does not lose any trades witha positive surplus while leading to a better price. In the N treatment, naive behavior is
not consistent with any equilibrium since the low-value buyer can obtain a higher expected
payoff by bidding equal to the highest ask strictly below v (which can be c). Similarly,
the high-cost seller would prefer to set ask equal the lowest bid strictly above c̄ (which canbe v̄). However, in the S treatment, naive behavior by traders with non-extreme values is
consistent with a fully eﬀicient equilibrium and thus not necessarily irrational. Naive play is
not a best response given players’ experience though: in the second part of the experiment(rounds 21-40) traders with non-extreme values play naively 29% of times, while in lessthan 5% of the times the best response to the observed history of play is within 0-4 distancefrom the cost/value.
One way to explain both the large proportion of naive play among non-extreme value
traders and the lower such proportion among extreme-value traders is that subjects have(non-monetary) costs of lying (or, more generally, disutility from deviating from cost/value).
To see this in the simplest possible way, consider an alternative model in which all players
are strategic but the buyer pays a cost of x if bid/ask is different from value/cost.38 Suppose
the low-cost seller asks z1 with c ≤ z1 < v while the high-cost seller asks z2 with c̄ ≤ z2 < v̄.
Then the low-value buyer gains 14 (v − z1 ) − x by setting bid equal z1 , whereas the high-value
buyer gains 41 (z2 − z1 ) − x. As the latter is greater than the former, we can thus havea situation where the low-value buyer is acting naively whereas the high-value buyer setsbid different from value – which is consistent with a greater proportion of naive play by
the traders with non-extreme values in the experiment. That the non-extreme value traders38

For example, Ellingsen and Johannesson (2004) and Serra-Garcia et al. (2013) consider models withsuch a fixed cost of lying.

25

have a lower expected gain from lying than the extreme-value traders could similarly explain
more naive play by the former but not by the latter types of traders in the DM frame if
framing the actions as a report about one’s cost/value increases the cost of lying.
Overall, past individual choices appear to influence present choices but the choices also
react to the previous experience in the manner one can expect, towards best responses toobservations. There is also some evidence that subjects may have disutility from deviationfrom cost/value. We use these results in the next section to motivate a dynamic model withreinforcement and noisy best response that can account for (some of) the properties of thedata from the experiment.

6 Behavioral Models of Individual Choices
Among the points learned from the previous sections are that the average behavior of theexperiment participants and eﬀiciency in the matches between human subjects are similar
across treatments S and N. This zero second-order effect is somewhat surprising because,
as discussed in Section 2, presence of naive traders dilutes the tradeoff between probability
and terms of trade, which could incentivize more strategic behavior in treatment N. The
average strategic behavior (for example, 25.54 for extreme and 4.26 for non-extreme value
traders in setting “20”) is also incompatible with playing a pure-strategy equilibrium witha small noise, hinting at the possibility that a non-equilibrium model may have a better fit
with data. Even though the average behavior and the eﬀiciency level hardly change overtime, individual behavior adjusts to some extent to the previous play. Finally, our findings
suggest subjects experience disutility from deviating from cost/value. In this section, we
introduce a dynamic model of behavior that reconciles these findings. We also compare thismodel to other models with a similar number of parameters, showing that it has a better
fit to the data.
We build the models on the discrete set of bids/asks as multiples of 5 between 0 and 100(the range used in the experiment): let J = {0, 5, . . . , 95, 100}. This set approximates thestrategy set actually used: even though in our experiment bids and asks could be decimal
numbers with up to two digits after the decimal point, 80% of choices were multiples of 5.39
39

Note that this discretization preserves at least one fully-eﬀicient equilibrium without naive traders in
our experimental treatments: equilibrium strategies Zsstr (c) = Zbstr (v) = 30 are in J, as well as Zsstr (c̄) =Zbstr (v̄) = 70.

26

6.1

A Dynamic Model

Consider the uniform strategy on J of one trader. Suppose that the choices of the othertrader follow a logistic distribution based on the expected payoffs against this uniform
strategy, i.e.,eλEu(j),(1)
P rob(j) = ∑λEu(i)i∈J ewhere j ∈ J and Eu(i) is the expected payoff of strategy i against the uniform strategy on
J. Parameter λ measures the noise in choosing the best response: for λ > 0, strategies withhigher payoffs have a higher probability to be chosen than strategies with lower payoffs; as
λ → ∞ the best response is chosen with probability approaching 1.40
The previous paragraph describes the choice of a subject with level-1 strategic sophistication (see, for example, Stahl and Wilson, 1994). We assume that this models the initialchoice of a strategy.41 To model the dynamics, we use the experience-weighted attraction(EWA) learning model (Camerer and Ho, 1999).42 In the EWA model, players form attractions for each strategy and then select a strategy according to the logistic equation(1) based on these attractions in place of expected payoff. Attractions are updated basedon experience, with those strategies that were played and those that either performed orwould have performed best receiving more weight. The EWA model thus combines featuresof both reinforcement and belief-based models of learning. Strategies that tended to do
better in past are reinforced and, at the same time, behavior is updated based on how wella strategy would have performed against the other players’ history of play. As discussed in
the previous section (Result 3), these properties of the adjustment of strategies appear to
be at least partially present in the data from our experiment.
More precisely, each strategy i ∈ J in the EWA model has an attraction Ait at the endof period t. These attractions are updated according toAit =φNt−1 Ait−1 + (δ + (1 − δ)Iit )πit
,
Nt

40

(2)

This logistic choice is used, for example, in the level-k model of choices (Stahl and Wilson, 1994), in
the logit quantal response equilibrium (QRE; McKelvey and Palfrey, 1995), and in the experience-weighted
attraction model (EWA; Camerer and Ho, 1999).
41
Although other specifications of initial behavior are possible, level-1 strategic sophistication is a reasonable compromise reflecting both the complexity of the game and individual incentives to get a good
payoff.
42
The EWA model has a better fit to observed behavior in a large class of games compared to both quantal
response equilibrium and other dynamic models (Camerer and Ho, 1999; Ho et al., 2007) and has been used
more recently to explain experimental data (e.g., Hong et al., 2021, and Masiliūnas, 2023).

27

where πit is the payoff that strategy i obtained (or would have obtained)43 against the
opponent’s strategy in round t, Iit = 1 if strategy i was used by the player at time t (andIit = 0 if it was not used), and φ, δ are parameters. The experience count Nt is updated
according to Nt = ρNt−1 + 1, where ρ and the initial experience count N0 are parameters.44
While the initial attractions Ai0 could also be treated as free parameters, as discussed above,
we set them equal to the expected payoff of strategy i against the uniform distribution of
opponent’s choice on J, modeling level-1 strategic sophistication in initial choices.
The model describes the attractions of strategies for a player who has to make one choice.
In our setting, each player can be in four situations: low- or high-cost seller, and low- andhigh-value buyer. Each player thus has four sets of attractions. Only the attractions for thecurrent role (buyer or seller) are updated in a given period by equation (2). For the current
role, both sets of attractions (for extreme and non-extreme cost/value) are updated (sincea player can use opponent’s strategy to evaluate what would have happened if cost/value
were different) but Iit = 1 only for the current cost/value (and actual strategy i played).
Motivated by the observed high frequency of “naive” behavior in the experiment, weconsider that the game payoffs also include disutility of deviating from cost/value, with
fixed component F (that applies only if ask/bid is different from cost/value) and marginalcomponent f for each unit of deviation from cost/value.45 Our model thus includes 7
parameters: 5 parameters (λ, δ, φ, ρ, N0 ) of the EWA model and 2 parameters (F, f ) of thedisutility of deviating from cost/value.
We estimate the model by maximum likelihood on the data from 10 sessions with setting “20” (we use 6 sessions with setting “25” for validation of the estimates, as described
below).46 The estimated parameters are λ = 0.231, δ = 0.525, φ = 0.848, ρ = 0.820, N0 =43

If payoffs in a game can be negative, then the interpretation of the term (δ + (1 − δ)Iit )πit as reinforcement is less clear (the multiplication of a negative payoff by a number less than 1 increases it, while such
multiplication for a positive payoff decreases it). We therefore add 70 to all payoffs in the game to makethem all non-negative. If initial attractions are based on expected payoffs, such an additive transformation
does not change the choice probabilities in equation (1) for round 1. For subsequent rounds it adds inertia
in the choice of strategies by increasing the reinforcement of the chosen strategy (Xie, 2021).
44
Parameter δ reflects the mix between reinforcement and belief-based learning. If δ = 0, then only thepreviously played strategy is reinforced, as in reinforcement learning, whereas if δ = 1, then all strategies
are considered for updating, as in belief-based learning. Parameters φ, ρ, and N0 are related to the speed oflearning, measuring the decay of previous attractions, the decay of experience and the strength of the priorrespectively. See Camerer and Ho (1999) for further explanation.
45
This form of deviation cost combines the forms of lying costs discussed, e.g., in Ellingsen and Johannesson(2004), Kartik (2009), Serra-Garcia et al. (2013) and Gneezy et al. (2018). Masiliūnas (2023) also considers
payoff modifications in EWA learning.
46
Details of the estimation procedure can be found in Section S4.1 of Supplementary Materials online.

28

Efficiency in setting "25"

Efficiency in setting "20"
1

1

0.95

0.95

0.9
0.8

0.9
0.85
0.8

0.75

0.75

0.85

0.7

0.7

0.65

0.65

0.6

0.6

0.55

0.55
0.5

0.5
1-5

6-10

11-15
S-Exp

16-20
N-Exp

21-25
S-Sim

26-30

31-35

1-5

36-40

6-10

11-15
S-Exp

N-Sim

16-20
N-Exp

21-25
S-Sim

26-30

31-35

36-40

N-Sim

Note: Blocks of periods on the horizontal axis; fraction of surplus realized on the vertical axis (expected eﬀiciency).
S: treatment S; N: treatment N. -Exp (thinner lines): experimental data; -Sim (thicker lines): simulated data.

Figure 4: Eﬀiciency in simulations and in actual sessions
4.692 for the EWA model parameters47 and F = 4.076, f = 0.081 for the disutility parameters, with the log-likelihood value LL = −11167.0.
We use these values of the parameters to simulate experimental sessions. Each simulated
run has a matching group of 8 subjects, randomly matched for 40 rounds and each using themodel above to make decisions. The average results of 100 simulated runs for treatments S
and N are presented in Figures 4 and 5.48 Thick lines are the averages from the simulations
while the experimental data are reproduced as thinner lines.
The average data from the simulations appear to fit the average data from the experiment quite well, both in terms of eﬀiciency measures and in terms of the extent of strategic
behavior. The realized eﬀiciency in treatment S in simulations is noticeably lower than the
expected eﬀiciency in treatment N, as in the experiment.49 There is little difference in theextent of strategic behavior between N and S treatments in simulations for both extreme
and non-extreme value traders. The simulated averages are also very close to those fromthe data – to the extent that some thin lines representing the experimental data are hardly
visible, being covered by the thicker lines of the simulated data. Even though the model is
47

The estimates are consistent with other estimates in the literature. For example, Stahl and Wilson(1994), McKelvey and Palfrey (1995), Camerer and Ho (1999), and, more recently, Nunnari and Zapal(2016), Hong et al. (2021), Masiliūnas (2023), estimate λ in range [0.1, 0.5] (adjusted for our range of
payoffs). Estimates for δ, φ and ρ are not far from those discussed in Camerer and Ho (1999). That δ is not
close to 0 or 1 indicates that both reinforcement and best-responding are important in the model.
48
In simulations of the N treatment, initial attractions based on the expected payoffs take into accountthat with probability 0.25 the opponent plays naively. Also, in each match, simulated subjects play naive
opponents with probability 0.25, as in the experiment. Attractions are still updated as described above.
49
The human matches eﬀiciency in treatment N in simulations (not shown to make the figure less cluttered)is only slightly lower than the realized eﬀiciency in treatment S.

29

Strategic behavior of extreme-value traders in "25"Strategic behavior of extreme-value traders in "20"
30

30

28

28

26

26

24

24

22

22

20

20
18

18
1-5

6-10

11-15
S-Exp

16-20
N-Exp

21-25

26-30

S-Sim

31-35

1-5

36-40

6-10

11-15
S-Exp

N-Sim

Strategic behavior of non-extreme value traders in "20"

16-20
N-Exp

21-25

26-30

S-Sim

31-35

36-40

N-Sim

Strategic behavior of non-extreme value traders in "25"

12

12

10

10

8

8

6

6

4

4

2

2

0

0
1-5

6-10

11-15
S-Exp

16-20
N-Exp

21-25
S-Sim

26-30

31-35

36-40

1-5

N-Sim

6-10

11-15
S-Exp

16-20
N-Exp

21-25

26-30

S-Sim

31-35

36-40

N-Sim

Note: Blocks of periods on the horizontal axis; average strategic behavior (v − b and a − c) on the vertical axis. S:
treatment S; N: treatment N. -Exp (thinner lines): experimental data; -Sim (thicker lines): simulated data. Top
panels: extreme values (c and v̄); bottom panels: non-extreme values (v and c̄).

Figure 5: Strategic behavior in simulations and in actual sessions
dynamic, there is little change in either eﬀiciency or average strategic behavior over time.
The parameters used in the simulations are estimated only on the data from setting“20”. The right panels of the figures provide a visual validation that the model with theseparameters also fits setting “25”: the averages of the simulated data are also close to theaverages of the experimental data in these panels. As a more formal validation measure
we use the mean absolute deviations between the simulated and experimental averages ofstrategic behavior for blocks of 5 periods used to draw Figure 5. For setting “20” in theleft panels, this measure is 1.260; for setting “25” in the right panels, the measure is 1.349.
That the measure is only 0.089 higher out-of-sample (while, as can be seen in the figure, theaverages of strategic behavior vary on the scale between 0 and 30) shows that the model
also works reasonably well for setting “25”, indicating little overfitting.50
50

As we discuss in the next subsection, the EWA model also has a better fit than other models, both inand out-of-sample.

30

6.2

Comparison with Other Models

Given that average strategic behavior changes little over time in the experiment, alternativemodels of behavior can assume that subjects use the same strategy in all periods.51
We first check whether a noisy equilibrium-like behavior can explain our data. For this,
we look at the actual distributions of behavior in each matching group.52 If behavior is close
to an equilibrium, it should fit with (possibly noisy, as represented by equation (1)) best
response to these distributions. We estimate such a model by maximum likelihood with
3 parameters: best response noise λ and two parameters F, f allowing for disutility fromdeviating from value/cost. The estimated maximum log-likelihood value is LL = −15502.1,
much lower than the value for the EWA model.
Constant average strategic behavior is also consistent with a model in which subjects
use a static strategy of setting ask somewhat above cost (or bid somewhat below value).
To allow for the observed persistence of individual behavior over time, we estimate such amodel where behavior of a subject, for a given type (extreme or non-extreme), comes fromone of two distributions over J (or from a mixture of these distributions). Each distribution
is based on a normal distribution with two parameters: mean strategy (modeling averagemark-up) and variance (modeling possible deviations from the average mark-up).53 Using
two distributions allows for subjects who may be naive, strategic, or a mixture of the two.
The maximized log-likelihood value of this model is LL = −14526.1, again worse than thevalue for the EWA model.
Thus, even though the average behavior over the 40 rounds of experiment changes little,
the dynamic strategy adjustment model fits the behavior better than the two static models.
While it is possible that some subjects do not change their behavior over time, suﬀiciently
many subjects modify their choices, justifying the consideration of the dynamic model.
In bilateral bargaining situations, players can also have preferences for a fair distributionof the trade surplus, with utility depending not only on player’ s own payoff but also on how
different it is from the other player’s payoff.54 In our setting, since a trader does not know
the cost or the valuation of the other trader, what is fair ex ante is not easy to determine
but a player can evaluate the fairness of an outcome ex post, if cost and value were observed
51

Details of the estimation procedures for the two static models below can be found in Section S4.2 of
Supplementary Materials online.
52
Since the double auction game has multiple equilibria, different matching groups can in principle beclose to different equilibria.
53
These normal distributions are adjusted to probability distributions on the discrete set J, as describedin Section S4.2 of Supplementary Materials online.
54
Roth (1995) discusses fairness in bargaining experiments; Fehr and Schmidt (1999) and Bolton andOckenfels (2000) are among the first to model such preferences.

31

In-sample (setting “20”)Out-of-sample (setting “25”)

Model

k

LL

AIC

BIC

LL

AIC

BIC

EWA (deviation cost)
EWA (fairness)“Equilibrium-like”“Mark-up”

7
8
3
12

−11167.0−11347.2−15502.1−14526.1

22348.022710.4
31010.1
29076.1

22395.822765.031030.629158.0−7119.8−7210.3−9972.3−8947.814253.6
14436.7
19950.6
17919.7

14301.3
14491.3
19971.1
18001.6

Note: k is the number of parameters in a model. n = 6800 observations for setting “20”, n = 4080 for setting “25”.
LL: log-likelihood on the parameters estimated for setting “20”; AIC: Akaike Information Criterion (2k − 2LL);
BIC: Bayesian Information Criterion (ln(n)k − 2LL). Models are as described in text.

Table 7: Comparison of models in- and out-of-sample
or can be inferred from ask/bid.55
We estimate a mixture EWA model, allowing a proportion of subjects to have fairness
preferences as in Fehr and Schmidt (1999). The maximized log-likelihood of this model is
LL = −11347.2, worse than the model with disutility of deviations from cost/value, eventhough the mixture model has one more parameter.56 In our experiment, the behaviorof extreme-value traders can be consistent with inequity aversion, but many non-extreme
traders make asks/bids close to their cost/value, which is diﬀicult to reconcile with fairness,since it leaves a large part of the surplus to the other trader. While it is possible thatsome subjects have fairness preferences, to fit the data such preferences would need to differ
depending on the position in the game (extreme or non-extreme). The model with disutilityof deviating from cost/value, with parameters that do not depend on the position in thegame, on the other hand, provides a more parsimonious explanation of behavior.
Table 7 summarizes the comparison of the models. Both the raw log-likelihood and thevalues of information criteria are better for the EWA model with disutility of deviating fromcost/value than for the alternative models. This holds for the sample used for estimation(data for setting “20”), as well as out-of-sample (data for setting “25”)57 , again confirmingthat the model, despite its large number of parameters, is not overfitted in-sample.
The comparison shows that the EWA learning model has the best fit among the considered models. There are still some differences between the data from EWA simulations andfrom experiment. For example, the behavior of non-extreme-value traders is more dispersed
55

Section S4.3 of Supplementary Materials online describes in more details how we model these preferencesin our setting.
56
The details of estimation are presented in Section S4.3 of Supplementary Materials online.
57
For the “mark-up” model the parameters estimated from setting “20” are modified for setting “25” byadjusting the mean strategy of the extreme value traders, as described in Section S4.2 of Supplementary
Materials online.

32

in the simulations than in the experiment, with such traders more often bidding above value
or asking below cost in simulations than in the experiment. The simulations have a build-in
symmetry across roles and do not distinguish between the BA and DM frames.58 Nevertheless, the simulations demonstrate that a learning model can explain the behavior in theexperiments to a large extent. In particular, the simulations show that with reinforcementof strategies and learning noisy best response, convergence to the eﬀicient equilibrium is
diﬀicult, and that the presence of naive traders does not necessarily lead to noticeably moreaggressive behavior and less eﬀiciency, confirming what we observe in our experiment.

7 Conclusion
In this paper, we analyzed a bargaining situation with incomplete information in which anatural double-auction mechanism has an eﬀicient equilibrium. However, with non-strategic
naive traders, who set bid/ask equal to value/cost, this eﬀicient equilibrium disappears;
thus, the presence of naive traders may decrease eﬀiciency.
We ran an experiment to see whether the presence of such naive traders would reduce
eﬀiciency but did not find this effect. We found that the experimental subjects did notplay more strategically after an introduction of artificial naive traders, and did not playaccording to the eﬀicient equilibrium in the treatment without such naive traders. Instead,subjects tended to reinforce strategies that worked well in the past as well as adjust behavior
toward strategies that would have done better against the history of other player’s play.
Motivated by this finding, we presented a learning model, with initial choices describedby noisy best responses according to low levels of strategic sophistication and subsequentadjustment based on past experience, that replicates average measures of eﬀiciency andbehavior from the experimental data (while also providing a better fit than other models
with a comparable number of parameters).
Taken at face value, our finding that play may be better described by learning startingfrom low levels of strategic sophistication appears to contradict the previous literature(Radner and Schotter, 1989; Valley et al., 2002; McGinn et al., 2003; Ellingsen et al., 2009).
That literature found that the constrained-eﬀicient linear equilibrium of the double auction
describes the data well even though the information setting in those experiments (uniform
distributions of value and cost on an interval) is more complex than our setting of twopoint value/cost distributions.59 There is one way to resolve these seemingly contradictory58

A possible way to get simulations have a better fit and reproduce observed differences across roles andframes is to have heterogeneity across subjects, with some having higher disutility from deviating fromvalue/cost or some never bidding above value or below cost.
59
Valley et al. (2002) and McGinn et al. (2003) find that pre-play communication generates eﬀiciency gains

33findings. In the model with identical uniform distributions of values, observed behavior
can be consistent with the linear equilibrium of the double auction even though players
have low levels of strategic sophistication and do not play the equilibrium strategies. For
instance, assuming uniform-random play at level-0, expected behavior is equal to the linear
equilibrium strategy if there are 25% of (non-noisy) level-1 players and 75% of (non-noisy)
level-2 players.60 Thus, it is possible that players’ behavior in that setting appears close
to linear equilibrium although it is better described by non-equilibrium models like level-k
and learning dynamics – of course, a rigorous reanalysis of data from previous experiments
is necessary to test this conjecture.
The conclusions reached in this paper, which contrast some of the previous literature,
call for further studies on private value bargaining. We need to understand better thenature of strategic behavior (whether it corresponds to equilibrium or to non-equilibrium
models, and whether and how its extent changes when parameters of the situation change),
as well as the possibility (or impossibility) of achieving full eﬀiciency in practice, also (andperhaps especially) when it is theoretically feasible in equilibrium.

Appendix: Proofs
Proposition 1. Suppose ε ∈ [0, 1). Strategies (Zsstr , Zbstr ) are a fully eﬀicient equilibrium
if and only if Zsstr (c) = Zbstr (v) = z1 and Zsstr (c̄) = Zbstr (v̄) = z2 , wheremax{A(c, v), B(z2 , c, v), C(z2 , c, v, v̄)} ≤ z1 ≤ D(c, v)(3)E(c̄, v̄) ≤ z2 ≤ min{F (c̄, v̄), G(z1 , c̄, v̄), H(z1 , c̄, v̄, c)}.

(4)

and

Proof. In an equilibrium in which a (strategic) seller with value c trades with a positive
probability, the ask cannot be strictly below the cost because the seller can get a higher
payoff by setting the ask equal to the cost. Similarly, the bid of a strategic buyer cannot
be strictly above the value. In a fully eﬀicient equilibrium, Zsstr (c) ≤ Zbstr (v) and Zsstr (c̄) ≤Zbstr (v̄). Thusc ≤ Zsstr (c) ≤ Zbstr (v) ≤ v < c̄ ≤ Zsstr (c̄) ≤ Zbstr (v̄) ≤ v̄.
that are greater than the theoretical maximum in the setting with uniform distributions. In Possajennikov
and Saran (2022), we show that allowing pre-play communication in our setting does not always lead to fulleﬀiciency.
60
For example, buyer’s level-1 and level-2 strategies are 2/3v and 2/3v +1/9, respectively. If 25% of buyers
are level-1 and the rest level-2, then the buyer’s expected behavior equals 2/3v + 1/12, which is the linear
equilibrium strategy of the buyer in the game. See Crawford (2021) for further details of level-1 and level-2
strategies in this setting.

34

If Zsstr (c) < Zbstr (v), then Zcstr (c) can be increased to Zbstr (b) increasing strategic seller’s
expected payoff. Thus Zsstr (c) = Zbstr (v) = z1 ≤ v and similarly Zsstr (c̄) = Zbstr (v̄) = z2 ≥ c̄.
The strategic seller with cost c gets in equilibrium the expected payoff
(
(
))
( (
)
(
))
1
1 z1 + z2
1 z1 + v
1 z1 + v̄
(1 − ε)(z1 − c) +
−c
+ε
−c +
−c
(5)
2
2
2
2
2
2
2
from setting ask equal z1 . From asking v the seller would get
))
( (
)
))
(
(
(
1 v+v
1 v + v̄1
1 v + z2+ε.
(1 − ε)
·0+
−c
−c +
−c
2
2
2
2
2
2
2

(6)

The difference between payoffs (5) and (6) is non-negative if and only if z1 ≥ A(c, v). The
seller would get the expected payoff
(
)
(
(
))
1
1
1
1 z2 + v̄
(1 − ε)· 0 + (z2 − c) + ε
·0+
−c
(7)
2
2
2
2
2
from setting ask equal z2 . The difference between payoffs (5) and (7) is non-negative if andonly if z1 ≥ B(z2 , c, v). Finally, from asking v̄ the seller would get
(
)
(
)
1
1
1
1
(1 − ε)·0+ ·0 +ε· 0 + (v̄ − c) .
(8)
2
2
2
2
The difference between payoffs (5) and (8) is non-negative if and only if z1 ≥ C(z2 , c, v, v̄).
All other deviations give a lower payoffs than one of the deviations considered above thus
z1 can be part of equilibrium if and only if z1 ≥ max{A(c, v), B(z2 , c, v), C(z2 , c, v, v̄)}.
The buyer with value v gets in equilibrium the expected payoff
(
)
( (
)
)
1
1
1
z1 + c1(1 − ε)(v − z1 ) + · 0 + ε
v−
+ ·0
(9)
2
2
2
2
2
from bidding z1 . Bidding c̄ and z2 cannot be profitable deviations. Bidding c would get thebuyer the expected payoff
)
(
)
(
1
1
1
1
·0+ ·0 +ε(v − c) + · 0 .
(10)(1 − ε)2222The difference in payoffs (9) and (10) is non-negative if and only if z1 ≤ D(c, v).
The second line of inequalities in the proposition follows analogously from considerationsof deviations from z2 .
If ε = 0, then A(c, v) = 31 (2c + v), B(z2 , c, v) = 13 (z2 + 2c) and C(z2 , c, v, v̄) = 13 (4c − z2 ).
For c̄ ≤ z2 ≤ v̄, max{A(c, v), B(z2 , c, v), C(z2 , c, v, v̄)} = 13 (z2 + 2c). Also, D(c, v) = v.
Therefore one inequality for fully eﬀicient equilibrium if ε = 0 is 23 c + 13 z2 ≤ z1 ≤ v.
Analogously, substituting ε = 0 into E(c̄, v̄), F (c̄, v̄), G(z1 , c̄, v̄) and H(z1 , c̄, v̄, c) givesc̄ ≤ z2 ≤ 32 v̄ + 13 z1 .
35

Proposition 2. Suppose that c = 10, c̄ = 70, v = 30, v̄ = 90.
i. If ε = 0, there exists a unique fully eﬀicient equilibrium Zsstr (10) = Zbstr (30) = 30 and
Zsstr (70) = Zbstr (90) = 70.
ii. If ε ∈ (0, 1), there is no fully eﬀicient equilibrium.
Proof. From Proposition 1, if ε = 0, the conditions for the existence of the fully eﬀicientequilibrium are 32 · 10 + 13 z2 ≤ z1 ≤ 30 and 70 ≤ z2 ≤ 23 · 90 + 31 z1 . The first condition canbe satisfied only if z2 ≤ 70. Then from the second condition z2 = 70 and, back from thefirst condition, z1 = 30. Thus Zsstr (10) = Zbstr (30) = 30 and Zsstr (70) = Zbstr (90) = 70 is
the unique eﬀicient equilibrium in this setting.
1
Consider now ε ∈ (0, 1). For the given parameter values, B(z2 , c, v) = 3−ε(z2 +20−30ε),1
1D(c, v) = 2−ε (60 − 50ε) and E(c̄, v̄) = 2−ε (140 − 50ε). Inequality (3) in Proposition
1 implies that B(z2 , 10, 30) ≤ D(10, 30), or that (2 − ε)z2 ≤ 140 − 130ε + 20ε2 . From
inequality (4), E(70, 90) ≤ z2 , or 140 − 50ε ≤ (2 − ε)z2 . The two inequalities imply that140 − 50ε ≤ 140 − 130ε + 20ε2 . This last inequality is equivalent to 20ε(4 − ε) ≤ 0, which
does not hold for any ε ∈ (0, 1). Therefore there are no values of z1 and z2 that satisfy
inequalities in Proposition 1.
Proposition 3. Suppose that c = 5, c̄ = 70, v = 30, v̄ = 95.
i. If ε = 0, there exist several fully eﬀicient equilibria.
√
ii. If ε ∈ (0, 1), a fully eﬀicient equilibrium exists for ε ∈ (0, ε̄], where ε̄ = (11− 101)/5 ≈0.190, and does not exist for ε ∈ (ε̄, 1).
Proof. For ε = 0, the conditions for the fully eﬀicient equilibrium from Proposition 1 are
1
2
1
2
3 · 5 + 3 z2 ≤ z1 ≤ 30 and 70 ≤ z2 ≤ 3 · 95 + 3 z1 . There are many values of z1 , z2 that satisfy
the two conditions, for example z1 = 30, z2 = 70 or z1 = 28, z2 = 72.
Consider now ε ∈ (0, 1). Inequality (3) in Proposition 1 means thatz1 ≤ D(c, v) =1
(60 − 55ε).
2−ε

(11)

Since G(z1 , c̄, v̄) = (z1 + 190 − 70ε)/(3 − ε), inequality (4), combined with (11) implies(3 − ε)(2 − ε)z2 ≤ 440 − 385ε + 70ε2 . Since E(c̄, v̄) = (140 − 45ε)/(2 − ε), inequality (4)also implies 140 − 45ε ≤ (2 − ε)z2 . Thus (3 − ε)(140 − 45ε) ≤ 440 − 385ε + 70ε2 , which is√equivalent to 0 ≤ 4 − 22ε + 5ε2 . This holds if ε ≤ ε̄ = (11 − 101)/5 ≈ 0.190 and does
not hold for ε > ε̄. Therefore for ε > ε̄ there is no z1 , z2 that satisfy the inequailities of
Proposition 1.
36

Take
z1 = D(5, 30) =1(60 − 55ε)
2−ε

and

z2 = E(70, 95) =1(140 − 45ε).
2−ε

(12)

By construction, z1 ≤ D(5, 30). It holds that A(5, 30) ≤ z1 if (2−ε)(40+20ε) ≤ (3−ε)(60−√55ε). This holds if ε ≤ 129/6 − 9/6 ≈ 0.393. For the given z2 , B(z2 , 5, 30) = (160 − 115ε +30ε2 )/((2 − ε)(3 − ε)). Then B(z2 , 5, 30) ≤ z1 if 32 − 23ε + 6ε2 ≤ (3 − ε)(12 − 11ε), or√ε ≤ (11− 101)/5 = ε̄. Finally, C(z2 , 5, 30, 95) = (−100+275ε−100ε2 )/((2−ε)(3−ε)) ≤ z1√
if −20+55ε−20ε2 ≤ (3−ε)(12−11ε). This holds if ε ≤ 50/31−2 191/31 ≈ 0.721. Therefore
for z1 and z2 in (12), inequalities in (3) of Proposition 1 are satisfied if ε ≤ ε̄. Analogous
reasoning shows that inequalities in (4) are also satisfied. Thus z1 and z2 in (12) constitutea fully eﬀicient equilibrium for any ε ≤ ε̄.

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