Discussion Paper No. 942

THE IMPACT OF MONITORINGIN INFINITELY REPEATED GAMES:PERFECT, PUBLIC, AND PRIVATE

Masaki Aoyagi
V. Bhaskar
Guillaume R. Frechette

July 2015

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

The Impact of Monitoring in Infinitely RepeatedGames: Perfect, Public, and Private ∗Masaki Aoyagi
Osaka University

V. Bhaskar
University of Texas at Austin

Guillaume R. Fréchette
New York University
July 27, 2015

Abstract
This paper uses a laboratory experiment to study the effect of a monitoring structure on the play of the infinitely repeated prisoner’s dilemma.
Keeping the stage game fixed, we examine the behavior of subjects when information about past actions is perfect (perfect monitoring), noisy but public(public monitoring), and noisy and private (private monitoring). We find thatthe subjects sustain cooperation in every treatment, but that their strategies
differ substantially in the three treatments. Specifically, we observe that thestrategies are more complex under public and private monitoring than under
perfect monitoring. We also find that the strategies under private monitoring
are more lenient than under perfect monitoring, and less forgiving than underpublic monitoring.
JEL classification: C72, C73, C92
Key words: Infinitely repeated games, monitoring, perfect, public, private, experiments∗
We wish to thank Matthew Embrey and Sevgi Yuksel as well as seminar participants at IAS
Princeton, Southern Methodist University, WZB, Stanford, Toulouse, Science Po, Harvard University, the University of Montréal, University of Heidelberg, the University of Munich, the Max-Plank
Institute, the University of Indiana, and conference participants at the North American Meeting of the
ESA, the Social Dilemmas conference, and the SITE conference for helpful comments and suggestions that improved the paper. We would further like to thank the National Science Foundation (grant
SES-SES-1225779), the CV Starr Center and the Center for Experimental Social Science (CESS) for
research support. We are responsible for all errors.

1

1

Introduction

Many economic situations involve repeated interactions among players who do notknow for sure what actions other players have chosen, or what information they
have observed about those actions. In a pioneering work, Stigler (1964) studies amodel of secret price cutting and points out the difficulty faced by the firms who
attempt to collude under (imperfect) private monitoring, where information about
the past actions (price choices) of other players is noisy and privately observed.
As Kandori (1992) notes, and by now is well recognized, private monitoring implies the absence of a coordination device for the players, and hence the lack ofa recursive structure in the repeated game. In this sense, there is a fundamental
difference between private monitoring and the more traditional environments ofperfect monitoring, where information about past actions is perfect (and public),and (imperfect) public monitoring, where the information is noisy but public. Theory shows that any equilibrium that sustains a positive degree of cooperation under
private monitoring must entail the play of an intricate mixed strategy.
A natural question then is whether people’s behavior under private monitoring is indeed significantly different from that under perfect or public monitoring.
Answering this question in the field is difficult because of data restriction: Private
information is unlikely to be replicated in publicly available data sets, and so is
information about other critical parameters such as the discount factor and the conditional distribution of signals given actions. In this light, a laboratory experimentoffers a valuable alternative to field research, and the objective of this paper is tocreate such environments in a laboratory and explore the behavior of subjects.
The first question this paper intends to explore concerns the ability of experimental subjects to sustain cooperation under private monitoring. Given the complexity of strategic interaction, it is only recently that theorists began to understandthe possibility of cooperation in a private monitoring environment. Hence, it would
not be surprising if the subjects fail to cooperate in a laboratory. The difficulty ofcooperation under private monitoring can be measured most accurately if we vary
the monitoring structures but keep the other aspects of the game fixed.
Our next question concerns if and how the subjects’ behavior under private
monitoring differs from that under perfect or public monitoring. We will answer
this question in two different ways. First, we examine if their behavior after certain
histories is different under different monitoring structures. Second, we estimate
their strategies and check whether the most popular strategies are different acrosstreatments. These analyses together help us answer the following questions among
others: How long back in history does a strategy look when choosing actions? Is it
lenient in the sense that it does not revert to a punishment after a single bad signal,or forgiving in the sense that it returns to cooperation after punishing the opponent?
2Our analysis is also an indirect test of the theory of private monitoring that implies
the use of extremely complex strategies required to sustain cooperation.
Our experiments use the repeated prisoners’ dilemma (PD) with a random termination probability of 0.1. There are three treatments corresponding to three different monitoring structures as follows. In the perfect monitoring treatment, player
i observes player j’s action choice atj in each round t. In the two imperfect monitoring treatments, on the other hand, player j’s action atj in each round t generates
signal ωti , which is correct and equals atj with probability 0.9, but is incorrect withprobability 0.1: In each round t, player i observes a pair of noisy signals (ωti , ωtj )in the public monitoring treatment, whereas he only observes ωti in the private
monitoring treatment.1 We keep fixed other elements of the game as much as possible: The three treatments have the same expected stage-payoffs, and the payoff in
the perfect monitoring treatment is determined randomly by the same probability
distribution as in the imperfect monitoring treatments to control for the effect of
uncertainty.
Our findings can be summarized as follows. First, we find that the rate of thecooperative action under private monitoring is comparable to those under perfectand public monitoring.2 Furthermore, the rate of coordination (either on (C, C) or(D, D)) is slightly lower under private monitoring than under perfect and public
monitoring, but significantly higher than implied by independent action choices.
These positive results on cooperation and coordination are remarkable in view of
the theoretical difficulties associated with private monitoring. Second, our strategy
estimation suggests that the subjects play substantially different strategies in thethree treatments. In particular, when we focus on the cooperative strategies that arefound in the most significant proportions in each treatment, none of them is lenient
under perfect monitoring, but all of them are lenient under public and private monitoring. As for forgiveness, the strategies used under private monitoring are not as
forgiving as those under public monitoring. Furthermore, when we represent thestrategies by finite automata and measure their complexity by the number of statesin this representation, we find the strategies under private monitoring more complex than those under perfect monitoring. These findings suggest that the subjects
find ways to cooperate and coordinate through a substantially different mechanism
under each monitoring structure.
The organization of the paper is as follows: In the next section, we give a brief
review of the literature. Section 3 formulates a model of repeated PD, Section 4
1

Hence, a player’s signal takes one of two values in the private monitoring treatment whereas it
takes one of four values in the public monitoring treatment.
2
In addition, we establish that their cooperation levels are higher than those in one-shot PD in theliterature.

3

provides a theoretical background, and Section 5 describes the experimental design. The questions our analysis attempts to answer are listed in Section 6, and theresults are presented in Section 7. Section 8 concludes with a discussion.

2

Related Literature

There is only indirect evidence from observational data as to whether repeatedinteractions under private monitoring lead to cooperation. A meta study by Levenstein & Suslow (2006) identifies the existence of a joint sales agency or industry
associations as mechanisms that help cartels through the collection and dissemination of information. Harrington & Skrzypacz (2007, 2011) find that cartels forsuch products as citric acid, lysine, and vitamins went to great lengths to makesales public information amongst members and also used inter-firm sales as a way
to transfer profits to sustain collusion. Their theory demonstrates that supporting
collusion requires side transfers in environments where information about prices isprivate. These findings indirectly attest to the difficulty of sustaining collusion justwith the help of private information.
As mentioned in the Introduction, the primary objective of this paper is to
identify the pure effect of the monitoring structure while keeping other aspects ofthe game fixed as much as possible. Although there is now a growing literature on
repeated game experiments, we are aware of no work that makes cross comparisonof different monitoring structures including private monitoring.3
Early experimental studies find some cooperation when subjects engage in repeated interactions under perfect monitoring.4 Further evidence of cooperation in
repeated games was provided by Engle-Warnick & Slonim (2004, 2006b,a), Dal Bó
(2005), Aoyagi & Fréchette (2009), and Duffy & Ochs (2009) in various settings(these subsequent studies differ from the earlier ones in that they allow subjects toplay multiple repeated games). Dal Bó & Fréchette (2011) find in perfect monitoring games that cooperation rates by experienced subjects are 1) very low whencooperation is theoretically infeasible, and 2) higher when it is theoretically feasible, and very high for certain parameter values. Furthermore, Dal Bó & Fréchette
3
Experiments on infinitely repeated games address a number of different questions. They include,
to mention a few, Schwartz et al. (2000), Dreber et al. (2008) on modified PD, Cason & Mui (2015) ona collective resistance game. Cooper & Kühn (2014) on the role of communication and renegotiation,
Fudenberg et al. (2014) on the relationship between behavior in the dictator game and that in an
infinitely repeated game, Cabral et al. (2014) on reciprocity, and Bernard et al. (2014) on a giftexchange game. Other forms of dynamic games are studied by Battaglini et al. (2015), and Vespa(2015).
4
See Roth (1995). Early studies include Roth & Murnighan (1978), Murnighan & Roth (1983),
Feinberg & Husted (1993), Holt (1985), and Palfrey & Rosenthal (1994).

4

(2013) find that in the repeated PD with perfect monitoring, the strategies used by
the majority of subjects are simple, and can be classified into one of 1) Always D(defect), 2) Grim Trigger, which begins with C (cooperate) but switches to D forever following a defection, and 3) Tit-For-Tat, which begins with C and thereafter
mimics the other player’s action in the previous round.
On games with imperfect public monitoring, Aoyagi & Fréchette (2009) findthat subjects support cooperation in a repeated PD with a noisy continuous signal,and that their payoff is a decreasing function of noise in the public signal in linewith the theoretical prediction on the maximal symmetric PPE payoff.5 Fudenberg
et al. (2012) study a model of repeated PD under imperfect public monitoring thatclosely resembles our public monitoring treatment. They formulate a model inwhich each player observes a profile of implemented actions that differ from theintended actions with positive probability, and examine the effects of stage payoffs
and noise levels on subjects’ behavior.6 Fudenberg et al. (2012) find no clear difference in the levels of cooperation caused by the monitoring structure, but that the
subjects’ strategies under public monitoring are more lenient and more forgiving
than under perfect monitoring. Using the strategy frequency estimation method
proposed in Dal Bó & Fréchette (2011), they support this finding by showing thatthe estimated strategies of the subjects choose actions based on the signals of thepast few rounds instead of the most recent one. In light of these results, it is particularly interesting to see what happens to leniency and forgiveness when monitoring
becomes private.7
To the best of our knowledge, the paper by Matsushima & Toyama (2013) isthe only other laboratory study of private monitoring in repeated games.8 They use
a repeated PD with private monitoring, and focus on how player behavior changes
in two treatments that change the noise level in the signal.9 As one would expect,
5Some, including Cason & Khan (1999), study repeated games with imperfect monitoring but do
not use random termination, which has become the standard procedure for implementing infinitely
repeated games in a laboratory since Roth & Murnighan (1978). See Fréchette & Yüksel (2013) forsome alternative termination methods and Sherstyuk et al. (2013) for alternative payment methods.
6
The payoff depends on one’s own choice and signal in our model whereas it depends on the two
implemented actions in Fudenberg et al. (2012).
7
Our analysis also provides a robustness check of Fudenberg et al. (2012), who use different
stage-payoff tables for different noise levels. In contrast, we keep the expected payoff table fixed
across treatments.
8
Holcomb & Nelson (1997) observe in a repeated duopoly model (without random termination)that the experimenter’s manipulation of information about a subject’s quantity choice “does significantly affect market outcomes” (p.79). Feinberg & Snyder (2002) also study the effect of occasional
manipulation of payoff numbers in a version of repeated PD, and find less collusive behavior whensuch manipulation is ex post not revealed than when it is.
9
The noise here refers to the probability that the signal differs from the actual action choice. In
our notation, see page 8, they have g = ` = 5/17 in their 0.9 treatment and g = ` = 5/19 in their 0.6

5they find that a player is more likely to cooperate when he observes a c (cooperative or good) signal rather than a d (defective or bad) signal in both treatments, andthat the cooperation rate is higher when the noise is smaller. The most interestingis their finding that players are more responsive to the signal when the noise issmaller. That is, the difference in the rates of choosing C after a c signal and aftera d signal is larger when the noise is smaller. This last finding is at odds with thetheoretical prediction based on a memory-one belief-free equilibrium.10 Our empirical findings point in the same direction with respect to responsiveness: players
are more responsive to the opponent’s signal under perfect monitoring than underpublic or private monitoring. We emphasize however that unlike in Matsushima &Toyama (2013), we don’t limit ourselves to memory-one strategies that condition
behavior only on the most recent signal.

1

Round 1
Fixed Pairs

Predicted Frequency of Cooperation
.2
.4
.6
.8

Groups of 60Groups of 14

1

2

3

4

5

6
7
Supergame

8

9

10

11

12

Figure 1: Duffy & Ochs (2009): random re-matching in small groups

Closely related to repeated games with private monitoring are models of random matching within a group where a group of players are matched in pairs to atreatment.
10
See Section 4 for the discussion of the belief-free equilibrium. Matsushima & Toyama (2013)
interpret these findings as a result of their subjects using a behavioral tit-for-tat strategy that incorporates psychological costs of cooperating and defecting.

6different partner every round. Monitoring is private – although a player perfectly
observes the action of his opponent in the current supergame, he does not observe
the actions taken in other pairs. Theoretically, regardless of the group size, cooperation can be sustained in equilibrium if δ is large enough through a contagious
grim-trigger strategy which cooperates as long as all past interactions have resultedin (C, C), but defects otherwise. Thus a single defection results in the breakdownof cooperation through the contagion process. Duffy & Ochs (2009) find that evenwhen the group size admits cooperation in theory, their subjects cannot sustain cooperation in the random matching environment. Their results are replicated in Figure 1. Subsequent experiments by Camera & Casari (2009), Camera et al. (2012),
and Camera & Casari (2011) also confirm that cooperation in the random rematching environment is fragile and is possible only for very small groups. The difficulty
to support cooperation with random rematching in small groups naturally poses aquestion as to whether players can sustain cooperation in bilateral interactions withprivate monitoring.

3

Models of Repeated Prisoners’ Dilemma

Two players play a symmetric 2 × 2 stage-game infinitely often. The set of actions
for each player i is denoted Ai = {C, D}. Player i’s action ai ∈ Ai generates a signal
ω j ∈ {c, d} with noise ε = 0.1. The probability distribution of ω j conditional on
ai is given by Pr(ω j = c | ai = C) = Pr(ω j = d | ai = D) = 1 − ε. The two
signals ω1 and ω2 are independent conditional on the action profile a = (a1 , a2 ):Pr(ω1 , ω2 | a) = Pr(ω1 | a2 ) Pr(ω2 | a1 ).
Under perfect monitoring, player i observes j’s action a j . Under (imperfect)public monitoring, player i observes the signal profile ω ≡ (ω1 , ω2 ). Under (imperfect) private monitoring, player i only observes ωi .11 With the identification
ωi = a j under perfect monitoring, player i’s stage-payoff is a function of his own
action ai and the signal ωi about player j’s action, and denoted by gi (ai , ωi ). Player
i’s expected stage-payoff ui is a function of the action profile a and is given byXui (a) =Pr(ωi | a j ) gi (ai , ωi ).
(1)
ωi ∈A jWe specify the function gi so that the expected stage payoffs (u1 , u2 ) form a PD as
11
Note that under private monitoring, there is no common knowledge between the two players
about each other’s action or signal. Furthermore, independence of ωi and ω j implies that i’s private
signal is uninformative about ω j under the unconditional choice of actions.

7follows:
C
C
1
D 1+g

D
1 −`
−` 0

1+g
0

(2)

In our experiments, the parameters g and ` > 0 are chosen to satisfy g = `.12 Let xit
denote player i’s information at the end of round t regarding events in the round as
described above.13 Player i’s history up to round t is the sequence hti = (xi1 , . . . , xit ).
t
Let Hit be the set of i’s histories up to t and let Hi = ∪∞t=1 Hi . Player i’s (behavioral)
1
t
t−1 →
strategy σi is a collection (σti )∞t=1 such that σi ∈ ∆Ai and for t ≥ 2, σi : Hi
∆Ai , where ∆Ai is the set of probability distributions over Ai . Denote by δ ∈ (0, 1)
the common discount factor of the players, and let πi (σ) be player i’s expected
payoff in the repeated game under the strategy profile σ = (σ1 , σ2 ). Likewise, let
πi (σ | hi ) be i’s expected continuation payoff under σ following history hi ∈ Hi .14
A strategy profile σ = (σ1 , σ2 ) is a perfect Bayesian equilibrium (PBE, or simply
an equilibrium) of the repeated game if for i = 1, 2,
πi (σ | hi ) ≥ πi (σ0i , σ j | hi )for any alternative strategy σ0i and any private history hi ∈ Hi .15 Under perfectmonitoring, σ is a PBE if and only if it is a subgame perfect equilibrium (SPE). Under public monitoring, a strategy σi is public if σti is a function only of (ω1 , . . . , ωt )and not that of (a1i , . . . , ati ). A PBE σ is a perfect public equilibrium (PPE) if each
σi is public.

4

Theoretical Background

This section collects some background material that is well recognized in the theoretical literature but is useful for the interpretation of our experimental results.
12
The equality g = ` implies that the expected payoff table has a benefit-cost form à la Fudenberg
et al. (2012). Namely, a player choosing C incurs cost g but gives benefit 1 + g to the other player,
whereas action D entails no cost or benefit. The same condition is referred to as separability in
Matsushima & Toyama (2013). The benefit-cost (b/c) ratio is given by 1+g.
g13
As mentioned earlier, the payoffs in the perfect monitoring treatment are randomly generated
according to the same distribution as in the imperfect monitoring treatments. As such, xit formally
includes the realization of the random payoff, which is by design independent of any player’s action
choice. See Section 5 for details of the actual implementation.
14
Throughout, we consider the average discounted payoff, which equals the sum of discounted
stage payoffs multiplied by 1 − δ.
15Under public and private monitoring, the definition of a PBE omits reference to the belief systemgiven the full-support assumption on the signal distribution Pr.

8

One essential observation concerns the players’ preference for efficiency/
cooperation and the severity of a punishment. Under perfect monitoring, cooperation in every period on the equilibrium path can be enforced by non-lenient andnon-forgiving strategies such as the grim-trigger strategy. Since no bad signal isobserved on the equilibrium path, leniency or forgiveness is immaterial for the efficiency of an outcome. Under imperfect public monitoring, on the other hand, badsignals arise even when both players cooperate. To achieve efficiency, hence, thestrategy must be lenient in the sense that a punishment is started either only after
the consecutive occurrence of bad signals, or with a small probability after each occurrence of such a signal. Furthermore, if the players are concerned with efficiency
a posteriori after the punishment is triggered, then the strategy must be forgiving
so that the cooperative phase can be restored after a fixed number of rounds orafter the occurrence of a good signal during the punishment phase. The situation
is significantly more complex with private monitoring. If player i believes that his
opponent j is playing a strategy that chooses C with probability one today but is
not lenient, then i’s strategy must be lenient: If i observes a bad signal today andresponds with D, then it will likely cause j to observe a bad signal and hence revert
to a punishment. On the other hand, if i is lenient and plays C instead, it will likely
keep j in the cooperative phase. After all, j does not know that i has observed abad signal, and since it is caused by the noise in monitoring, i might as well ignore
it. This reasoning excludes the possibility of an equilibrium that entails the unconditional play of C on the path along with a non-lenient response to a bad signal.
Theory of private monitoring suggests that j’s strategy must be finely adjusted in
the level of leniency and forgiveness so that i has an incentive to play C after agood signal and D after a bad signal.
A more specific description of an equilibrium in each case is as follows. Undergperfect monitoring, mutual cooperation is an SPE outcome if δ ≥ 1+g. For example, if we denote by CC the action-signal pair (ai , a j ) = (C, C), the grim-trigger
strategy σG that begins with ai = C and plays C if hti = (CC, . . . , CC) but plays
D otherwise, is an equilibrium strategy and generates the maximum symmetricaverage discounted payoff of 1.
Under public monitoring, a pair (σG , σG ) of grim-trigger strategies that revert
to punishment when the history hti , (Ccc, . . . , Ccc) is also a PPE for δ sufficiently
large and ε sufficiently small.16 However, such an equilibrium entails a significant
efficiency loss since permanent defection is triggered with probability 1 − (1 − ε)2
in every round.17 We can verify that the use of a lenient strategy that triggers a
As before, (ai , ωi , ω j ) = (C, c, c) is abbreviated as Ccc. (σG , σG ) is a PPE if (1 − 2ε)(1 − ε) −ε(2 − ε)g ≥ 1−δ
g.
δ
1−δ
17
The equilibrium expected payoff under this grim-trigger strategy equals 1−δ+δε(2−ε), which equals
0.369  1 under our parameter values (δ = 0.9 and ε = 0.1).
16

9

punishment with a positive but small probability after a bad signal yields a more
efficient equilibrium.
In the case of private monitoring, the lack of common knowledge of histories
becomes a major obstacle for cooperation. In particular, an equilibrium should bein mixed strategies if it supports any level of cooperation. As mentioned in the
Introduction, two approaches to the problem have been developed in the literature
as illustrated below.
The belief-based approach attempts to provide a proper incentive after each history by considering a mixture of repeated game strategies. Specifically, consider amixture between the grim-trigger strategy σG and the strategy σD of choosing D
always. Note that the continuation strategy of such a mixed strategy after each history is either σD or again a mixture of σG and σD . The initial probability weights
on σG and σD are chosen so that after every history, it is incentive compatible torevert to σD if and only if a player observes a d signal.18 In one interpretation,
when players are randomly matched to play the repeated game as in our experimental setting, a mixed strategy played by a single opponent corresponds to the
population of opponents playing different pure strategies.
Another approach to private monitoring is belief-free equilibria in which eachplayer plays a behavior strategy that makes the other player indifferent between C
and D after every history. Specifically, player i’s strategy makes player j indifferent between his actions independent of the history observed by player j. This
makes player j’s belief about player i’s (private) history irrelevant, and substantially simplifies the equilibrium analysis. A belief-free equilibrium constructed by
Ely & Välimäki (2002) can be illustrated as follows:19 Let p be the probability that
player i plays C when he observes signal ωi = c in the previous round, and q be
the same probability when he observes ωi = d. Furthermore, let W(c) denote i’s
continuation payoff when his opponent j observes ω j = c, and W(d) denote his
continuation payoff when j observes ω j = d. Since player i is assumed indifferent
between playing C and D, we should have
(1 − δ)g = δ(1 − 2ε) [W(c) − W(d)] ,(3)

where the left-hand side is i’s payoff gain in the current round from playing D rather
than C, and the right-hand side is the increase in continuation payoff from playing
C rather than D, which increases the probability of player j observing ω j = c by18

It is typically the case that with high discount factors, the players do not have an incentive toswitch to σD when observing ωi = d. This is the case with our specification of δ = 0.9, and it isnecessary to lower the effective discount factor by partitioning the supergame into several segments
so that each segment is played only once in several rounds. See for example Sekiguchi (1997).
19
See also Piccione (2002) for an alternative formulation of belief-free equilibria.

101 − 2ε.20 Next, if player j observed ω j = c in round t − 1 and player i plays D inround t, then player i’s continuation payoff from round t on is given byW(c) = (1 − δ) {p(1 + g) + (1 − p) · 0} + δ {(1 − ε)W(d) + εW(c)} .
On the other hand, if player j observed ω j = d in round t − 1 and player i plays D
in round t, then player i’s continuation payoff from round t on is given byW(d) = (1 − δ) {q(1 + g) + (1 − q) · 0} + δ {(1 − ε)W(d) + εW(c)} .
These equations together imply
W(c) − W(d) = (1 − δ)(p − q)(1 + g).

(4)

Combining (3) and (4), we obtain
p−q=

g
.
δ(1 − 2ε)(1 + g)

(5)(5) provides one behavioral prediction when the subjects play belief-free equilibrium that has memory-one in the sense that mixed actions are determined by thesignal realization of the previous round. Let the responsiveness of a strategy be
defined by= C | ωti = c) − Pr(at+1= C | ωti = d).
(6)Pr(at+1iiWhen the subjects play the memory-one belief-free equilibrium described above,
this quantity equals p − q and is expressed in terms of the underlying parameters
as in (5).
It can also be verified that the above belief-free strategy profile is an equilibrium not only under private monitoring but also under perfect and public monitoring. Consequently, if the subjects play the memory-one belief-free equilibrium inevery monitoring treatment, then they should exhibit the same responsiveness inboth the public and private monitoring treatments where ε = 0.1, and a lower responsiveness value in the perfect monitoring treatment where ε can be interpreted
as 0.

5

Experimental Design

The experiment has three treatments corresponding to the three monitoring structures described above. The public and private monitoring treatments use the payoff
20

Note that the gain from playing D does not depend on j’s action when g = ` as assumed.

11

Table 1: Summary statistics
Treatments
Perfect
PublicPrivateSubjects

Sessions

66
68
72

4
4
4

Subjects
per session
16, 18, 16, 16
18, 20, 14, 16
20, 18, 14, 20

Supergames
per session
11, 12, 19, 11
11, 11, 22, 11
12, 9, 19, 11

Rounds per Supergame
avg min
max
10.3
1
37
10.1
1
37
10.4
1
37

Subject earnings (US$)
avg
min
max
32.91 19.78 43.67
34.87 23.40 48.12
31.92 22.34 44.12

function gi (ai , ωi ) given byai \ωi
C
D

c
46
54

d
8
16

(7)

The expected payoffs are then generated according to (1). Our perfect monitoring
treatment introduces the same random relationship between the payoffs and theaction profile as follows: For each action profile (ai , a j ), player i’s payoff in theperfect monitoring treatment is generated by the lottery that yields gi (ai , ωi = a j )with probability 1 − ε and gi (ai , ωi , a j ) with probability ε. For example, when theaction profile is (C, C), each subject (independently) receives 46 with probability
1 − ε and 8 with probability ε so thatui (C, C) = (1 − ε) gi (C, c) + ε gi (C, d),just like in the other two treatments. It follows that our three treatments haveexactly the same expected stage-payoff table. With our choice of ε = 0.1, it isgiven bya1 \a2CDC
42.2, 42.2 11.8, 50.2
(8)D
50.2, 11.8 19.8, 19.8
Note that the payoff matrix (8) is strategically equivalent to (2) forg=`=5≈ 0.357.21
14(9)

In each of the three treatments, these parameter values ensure that the type of strategy discussed in the previous section is an equilibrium in which the players cooperate with strictly positive probability at least initially.
The experiments use the between-subject design so that each subject participates in one and only one treatment. Sessions were conducted at the CESS lab at
Simply apply the affine transformation 22.4 ui (a) + 19.8. The benefit-cost ratio mentioned in= 3.8.
Footnote 12 hence equals 1+g
g
21

12

NYU.22 In each session, after the instructions are read aloud, subjects are randomly
and anonymously paired via computer with another subject to play a supergame.23
All supergames in a session are simultaneously terminated after every round withprobability 0.1, and subjects are randomly rematched to play another supergame.24
After every supergame, subjects are informed of the complete history of choices
and signals by both players to ensure that feedback is the same for all treatments
and that the only difference among them is the information structure within a supergame. This process repeats itself until 75 minutes of play have elapsed; thefirst supergame to end after that marks the end of a session. Four sessions of eachtreatment were conducted. The supergames lasted between 1 and 37 rounds, andaveraged 10.3 rounds (close to the expected value of 10).25 The sessions wereapproximately 1 hour and 40 minutes, and subjects earned between $19.78 and$48.12 with an average earning of $33.21.26 These and other summary statistics
are provided in Table 1.27

6

Directions of Analysis

As discussed in the Introduction, the primary focus of our research is on (1) whether
subjects can support cooperation under private monitoring and how the level ofcooperation compares with those under public and perfect monitoring, and (2)whether or not their behavior is different in the three treatments. While our investigations are more of exploratory nature rather than hypothesis-testing of theoreticalpredictions, we will relate the results to insights and predictions provided by the22
Subjects who had participated in previous experiments with randomly terminated games or a PD
as a stage-game were excluded.
23
In the experimental instructions, the term “match” is used in place of “supergame.”
24
The length of a supergame in each session of the perfect monitoring treatment was determinedby a random number generator. Each session in the other treatments then used the same sequence ofsupergames as the corresponding session in the perfect monitoring treatment to control for the effectof the length of supergames on the evolution of play. Dal Bó & Fréchette (2011) and Engle-Warnick& Slonim (2006b) both document the impact of the length of supergames on behavior.
25
The difference in the average number of rounds results from the variation in number of supergames between sessions.
26
Points are converted to dollars at a pre-announced exchanged rate. Since the earnings for the
first session of each treatment were slightly lower than expected (between $19.78 and $33.52 with anaverage of $28.64), the minimum time of play was increased from 60 to 75 minutes and the exchange
rate was decreased from 0.01 to 0.0075 for the subsequent sessions.
27
Given the difference in the number of supergames across treatments, the analysis uses data fromonly the first kn supergames in session n of each treatment, where kn is the minimal number ofsupergames in session n across treatments. For instance, since the second sessions of the threetreatments have 9, 11 and 12 supergames, only the first 9 supergames are used in the analysis. Recall
that the length of the kth supergame in session n is the same regardless of the treatments.

13

theory to the extent possible. In what follows we provide a set of questions thatprovide a guide for our analysis.
Question 1 (Cooperation and coordination) Is the level of cooperation and coordination lower under private monitoring than under perfect or public monitoring?
Theory clearly suggests that it is significantly more difficult to sustain cooperation under private monitoring since it requires the use of intricate mixed strategies
as seen in Section 4. The lack of a coordination device under private monitoringalso makes it difficult for the subjects to coordinate their actions beyond round 1.
These considerations suggest an affirmative answer to question 1.
We examine our second question on the constancy of behavior across the threetreatments from several different perspectives. Following the literature and as described earlier, we say that strategies are lenient if they do not prescribe sure defection following a single bad signal, and forgiving if they return to cooperation after
having played defect.
Question 2 (Leniency and forgiveness) Are strategies more lenient and forgiving
under public and private monitoring than under perfect monitoring?
As mentioned in Section 4, a player’s preference for efficiency implies that his
strategy under public monitoring should be more lenient and forgiving than under perfect monitoring. Indeed, previous work confirms this view. In the perfectmonitoring environment, Dal Bó & Fréchette (2013) find grim trigger, which is notlenient, among one of the three most frequently observed strategies. To the contrary, both Fudenberg et al. (2012) and Embrey et al. (2013) find that the subjects’strategies are more lenient and more forgiving in the imperfect public monitoring environments. A similar observation can be made on the strategies that best
describe the subjects’ behavior in Aoyagi & Fréchette (2009): as noise in public
information increases, the range of a “bad” signal which causes transition fromthe cooperation phase to the punishment phase shrinks, and the range of a “good”
signal which causes transition from the punishment phase to the cooperation phasewidens.28
Under private monitoring, on the other hand, a strategy may be more lenient
than under perfect monitoring if a player believes that his opponent plays C withhigh probability: Because of the noise in monitoring, he will give a benefit of doubt
when he observes a single bad signal, and will also be reluctant to punish it since it
28

The signal space is continuous and the estimated strategies shift between the cooperation andpunishment phases based on a threshold on the public signal. Data of Aoyagi & Fréchette (2009)
show that this threshold decreases as noise increases.

14

will more likely invoke an adverse reaction from the opponent. The answer to thequestion on forgiveness under private monitoring is more difficult to predict sinceforgiveness, if any, takes place further down the history where it is difficult to infer
if the opponent is cooperative or punitive.
Question 3 (Memory length) Do strategies have longer memory under public andprivate monitoring than under perfect monitoring?
An affirmative answer to this question is partially implied by the affirmative answers to Question 2 on leniency since leniency requires the examination of history
over the past few rounds rather than just one.29
Question 4 (Responsiveness) Is the level of responsiveness lowest under perfectmonitoring and the same under public and private monitoring?
A belief-free strategy profile as described in Section 4 is an equilibrium inevery monitoring treatment we consider. The memory-one belief-free equilibriumis the simplest of them, and if the subjects indeed play such an equilibrium, theresponsiveness defined in (6) should be as described above.

7

Results

We present our results in two parts. The first part is a direct analysis of cooperationand coordination rates as well as action choices conditional on some histories. The
second part is an analysis based on the estimation of strategies.

7.1

Cooperation Rates

Cooperation rates in the three treatments can be assessed visually in the left panelof Figure 2. In light of the variation in the number of supergames across sessions,the figure presents data in three categories: the first four supergames to the left, thelast four supergames to the right, and a single point in the middle (labeled “other”)
that corresponds to the average of the rates in all other supergames. As such, every
point in Figure 2 (with the exception of the middle point) represents the average offour supergames, one from each session.
Observation 1 Subjects support cooperation under perfect, public and private monitoring.
29

Leniency can be defined in terms of either the number of bad signals before action D is chosen,or the probability with which action C is chosen after each bad signal. In the latter case, moreleniency does not necessarily imply longer memory.

15

All Rounds
11Round 1
Perfect
Public.8Frequency of Cooperation
.4
.6
.2
0

0

.2

Frequency of Cooperation
.4
.6

.8

Private1

3
2

Others
L−2
Last
4
L−3
L−1
Supergame

1

3
2

Others
L−2
Last
4
L−3
L−1
Supergame

Figure 2: Cooperation rates by supergame

Observation 1 on perfect monitoring replicates earlier results in the literatureand extends them to the environment with random payoffs. The round 1 cooperation rate in the last four supergames is 65%, which is statistically different from 0
at the 1% level.30
Although there is no direct comparison available, the level of cooperation observed under perfect monitoring is considered high even when we use as a benchmark the level of cooperation in one-shot PD. In fact, the finding of Dal Bó &Fréchette (2013) suggests that the level of cooperation observed here can be sustained only if the discount factor (continuation probability) is higher than the critical level required for the existence of a cooperative equilibrium.31
30

Throughout the paper, unless stated otherwise, statistical tests are obtained by t-tests clustering
the standard errors by session using only the last four supergames. The clustering is to account forpotential session-effects. The interested reader is referred to Fréchette (2012). When results arereferred to as not statistically significant, it implies a p-value greater than 10%.
31
See the Appendix A.1, which replicates the finding of Dal Bó & Fréchette (2013). In theirmodel, cooperation is a subgame perfect equilibrium outcome if and only if δ ≥ 0.72. As seen in
Figure 6, cooperation rates for δ = 0.9 and δ = 0.5 diverge as the subjects accumulate experience.

16

Recall that in the present perfect monitoring treatment, payoffs are randomly
generated conditional on the pair of action choices. In contrast, in the perfect monitoring repeated PD of Rand et al. (2015), action choices are implemented with error
although the intended action choice is observed. The cooperation rates observed
here are similar to those found in Rand et al. (2015), suggesting the robustness ofsubject behavior with respect to the way randomness is introduced.
Turning now to public monitoring, we see in the left panel of Figure 2 that theround 1 cooperation rate in the last four supergames is 73%, which is again statistically different from 0 at the 1% level. Positive cooperation in our experiments
is in line with the findings in the literature on various forms of public monitoring:
Fudenberg et al. (2012) introduces implementation errors in the subjects’ action
choices unlike in our experiment where noise is in the observation of the otherplayer’s action choice. In Aoyagi & Fréchette (2009), the subjects’ action choices
give rise to a one-dimensional continuous signal with infinite support that does notstatistically identify the deviator, for example, from the action profile (C, C). In
Embrey et al. (2013), outcomes depend probabilistically on the subjects’ action
choices as in Fudenberg et al. (2012), but the public signal is binary.
Between perfect and public monitoring, there is no statistical difference in theround 1 cooperation rates. As depicted in the right panel of Figure 2, the cooperation rates over all rounds in the last four supergames are 46% under perfectmonitoring and 58% under public monitoring. The rates are both statistically different from zero (p < 0.01), and statistically different from each other (p < 0.01).
Comparison of overall cooperation rates under public and perfect monitoring in theliterature is not conclusive, and suggests the influence of the specific monitoring
technology used. A finding similar to the one above is reported by Fudenberg et al.
(2012), who observe a statistically significant increase in overall cooperation rates
when small noise is introduced into monitoring. On the other hand, movementin the opposite direction with the introduction of small noise into monitoring isreported by Aoyagi & Fréchette (2009).32
The key finding in Observation 1 is cooperation under private monitoring. We
see again in the left panel of Figure 2 that in the last four supergames, there is 61%
cooperation in round 1, which is statistically different from 0 at the 1% level. The
cross comparison across the treatments reveals that the only difference between
round 1 cooperation rates is between public and private.33 In particular, the rates
32
In fact, Aoyagi & Fréchette (2009) report a monotonically decreasing relationship between noiseand cooperation rates in all rounds. They observe no statistical difference in round 1 cooperation rates
in the treatments where cooperation is theoretically feasible.
33
In addition to the description in footnote 30, statistical tests involving comparisons across treatments control for the random sequence of supergames. This is done to take into account the potential
correlations due to the fact that the realized lengths of supergames has been shown to affect choices

17are not statistically different between perfect and private. The cooperation rates
over all rounds in the last four supergames are 46% under private monitoring as
depicted in the right panel of Figure 2. It is again statistically different from zero(p < 0.01). There is no statistical difference between perfect (46% cooperation)and private, or between public (58% cooperation) and private. These observations
are summarized below:
Observation 2 Whether in round one or all rounds, the cooperation rates under
private monitoring are not different from those under perfect monitoring. The rates
in all rounds under private monitoring are not different from those under public
monitoring.
We should emphasize that Observation 2 is remarkable considering the theoretical difficulty in supporting cooperation under private monitoring. Again, a usefulcomparison is with the level of cooperation in one-shot PD. Dal Bó & Fréchette(2014) assemble a data set of 122,233 choices in eleven infinitely repeated andone-shot prisoner’s dilemma experiments. They run a probit regression of roundone cooperation rates on the payoff parameters g and `, discount factor δ, and indicators for subgame perfection and risk dominance.34 Using this regression, we canpredict the level of cooperation in one-shot PD under our parametrization. We findthat the observed round one cooperation rate in the private monitoring treatment
is higher by 26% (significant) than the prediction. A similar regression using only
data from four one-shot PD experiments with 11,038 choices also shows that theround one cooperation rate in our private monitoring treatment is higher by 35%(significant) than the predicted value. This shows that Observation 2 cannot be
simply explained by the behavioral hypotheses often used to explain cooperationin one-shot PD.

7.2

Coordination

As mentioned previously, the critical feature of private monitoring is the lack of acoordination device. In theory, players can perfectly coordinate their actions under both perfect and public monitoring, but not under private monitoring. In thissense, it is interesting to see if the subjects indeed have difficulty coordinating theiractions under private monitoring. Figure 3 extracts the first five rounds of eachsupergame in the three treatments and presents the values of Pr at = (C, C) and
Pr at = (D, D) as well as their sum. It also depicts the values of Pr(ati = C)2 andPr(ati = D)2 , which would be the coordination rates should the subjects choose their(see for instance Dal Bó & Fréchette (2011)).
34
This is estimated supergame by supergame.

18

Public

Private

.6
.4
0

.2

Coordination Rate

.8

Perfect

1

2

3

4

5

1

2

3

4

5

1

2

3

4

5

Round
Exp. DExp. C

Coord. D

Coord. C

Total* Last 4 Matches

Figure 3: Coordination rates: Implied by independent action choice and realized
actions independently. Coordination rates Pr at = (C, C) or (D, D) are 0.722 and0.712 under perfect and public monitoring, respectively, and no statistical difference exists between them. In comparison, the coordination rates are 0.660 under
private monitoring, and there is a statistical difference between public and private. However, the difference is relatively small and it is surprising to see howmuch coordination is achieved under private monitoring after the initial round
despite the difficulty implied by the theory. When we compare Pr at = (C, C)
and Pr at = (D, D) with Pr(ati = C)2 and Pr(ati = D)2 , respectively, we find thatthe former is always higher except in the first round. In fact, the coordination

rates Pr at = (C, C) or (D, D) are higher by 13 percentage points than Pr(ati =C)2 + Pr(ati = D)2 for all rounds after round 1, corresponding to a 41% difference
for perfect and 84% for public. For private, the rates are higher by 12 percentage
points, corresponding to a 40% difference. This suggests that, to a certain extent,
subjects have the correct expectation about the other player’s action.
Observation 3 Coordination rates under private monitoring are (insignificantly)

19lower than those under perfect and public monitoring, but are positive and significantly higher than the level implied by independent action choices.

7.3

Conditional Cooperation

PublicPrivate1

11

Perfect

t=1

ωi =c
t−1

ωi =d

1
2

3
O L−2 Last
4 L−3 L−1
Supergame

.8
0

.2

.4

.6

.8
.2
0

0

.2

.4

.6

Frequency of Cooperation
.4
.6

.8

t−1

1

3
2

O L−2 Last
4 L−3 L−1
Supergame

1

3
2

O L−2 Last
4 L−3 L−1
Supergame

Figure 4: Cooperation conditional on the previous signal
In this subsection we explore further how cooperation rates vary with the actionsignal pair in the preceding round in each monitoring environment. We begin by focussing on cooperation rates conditional only on the signal in the preceding round.
Figure 4 shows the rates with which player i chooses ati = C in round t ≥ 2 when
his signal in round t − 1 is c (labeled ωt−1= c), when it is d (labeled ωt−1
= d),
i
i
and when t = 1 (labeled t = 1). Clearly, cooperation rates following a good signal
are much higher than following a bad one (p < 0.01 in all cases). Another striking
point is that this difference increases as the subjects accumulate experience. For
instance, in the first supergame, the difference in cooperation rates following thetwo signals is between 23 and 26 percentage points in any treatment, whereas in
the last supergame, the corresponding difference is 59 percentage points under per20

fect monitoring, 51 percentage points under public monitoring, and 54 percentage
points under private monitoring.
Observation 4 In every monitoring treatment, the rate of the cooperative action C
is higher after a good signal c about the opponent’s action than after a bad signal dabout it.
Figure 4 also shows that (1) responsiveness (= difference in cooperation rates
following a c signal and a d signal defined by (6)) varies across treatments, and that(2) round one cooperation rates are about the same as cooperation rates following agood signal under perfect and public monitoring, whereas they are different under
private monitoring. If we suppose that the subjects play the memory-one belieffree equilibrium described in Section 4, then responsiveness should in theory belowest under perfect monitoring at approximately 0.292, and about 0.365 under either public or private monitoring. Our data show, however, that responsiveness under perfect monitoring is higher than that under public or private monitoring: The
numbers are 0.354, 0.249, and 0.295 for the perfect, public, and private treatments,
respectively.35 A joint test reveals that responsiveness in the perfect monitoring
treatment is statistically different (p < 0.05) from that in the other two. Comparedwith the theoretical prediction in each case, the observed responsiveness is significantly different (p < 0.01) in the private monitoring treatment, different but not assignificantly (p=0.094) in the public monitoring treatment, and not different in theperfect monitoring treatment.36 We also note that responsiveness is rather sensitive to the choice of a specific sample. For instance, although the same predictions
should apply to all rounds after the first, if we compute responsiveness in round2 only, it is 0.181, 0.236, and 0.305 for perfect, public, and private monitoring,
respectively. Notice that responsiveness is now lower under perfect monitoring
than under public and private monitoring (not statistically different however). The
observed discrepancy from the theoretical prediction based on the memory-one
belief-free equilibrium may come from a number of different sources. One important consideration is that strategies condition on events beyond the most recent
signal. This point is examined in more detail later.
While the above analysis only considers the action choice conditional on themost recent signal, it may as well depend on one’s own action in the previous
round. The relationship between the action choice in the present round and one’s
35

These numbers are obtained by first computing responsiveness for each subject and then performing t-tests on these subject averages with clustering at the session level.
36
Responsiveness is significantly higher under perfect monitoring than under public monitoring(p < 0.01), and (insignificantly) higher under perfect monitoring than under private monitoring(p = 0.104). Furthermore, the levels are statistically different between public and private monitoring(p < 0.05).

21



Pr ati = C | Cc
Pr ati = C | Cd
Pr ati = C | Dc
Pr ati = C | Dd

perfect
0.946
0.414
0.223
0.114

public
0.922
0.553
0.470
0.135

private
0.921
0.469
0.351
0.105

Table 2: Signal and cooperation rate conditional on previous choice and signal
combination
own action choice and signal in the previous round is summarized in the Table 2,t−1
where (at−1i , ωi ) = (C, c) is abbreviated as Cc, etc.
First note that cooperation rates across treatments are about the same (jointly
not statistically different) when the subjects previously cooperated and received agood signal (the first row) and when they previously defected and received a badsignal (the fourth row). The main differences are in the cooperation rates after
Cd and Dc. Note that the higher cooperation rate after Cd implies more leniency,
whereas the higher rate after Dc corresponds to more forgiveness. With this interpretation, strategies under public monitoring exhibit more leniency and forgivenessthan those under perfect monitoring (both are statistically different), and strategies
under private monitoring come somewhere in between in both dimensions. We
should note, however, that this is only a rough measure of leniency and forgivenessas the strategies may condition on events beyond the previous round. More direct
analysis of leniency and forgiveness is performed in conjunction with the estimation of strategies. Table 2 also shows that subjects under private monitoring react
to a negative signal more strongly when they cooperated: the difference between
the cooperation rates after Cc and after Cd is the largest in this treatment.
Observation 5 In every treatment, the rates of cooperative action C substantiallyvary with the signal in the previous round as well as the action-signal pair in theprevious round. The cooperation rates after Cd and Dc are the highest under public
monitoring and lowest under perfect monitoring.

7.4

Leniency and ForgivenessTo further investigate the question of leniency and forgiveness, we study behavior
after some key histories that are possibly longer than one round.37
37

See Fudenberg et al. (2012) for a similar exercise.

22

Leniency
11Forgiveness
Perfect
Public.8Frequency of Cooperation
.4
.6
.2
0

0

.2

Frequency of Cooperation
.4
.6

.8

Private01
2Consecutive d Signals / No previous D Sequence0

1
2
Consecutive c Signals / First D Sequence* Last 4 Matches
Rounds > 2
Right panel excludes matches that start with defection

Figure 5: Cooperation rates after d signals when cooperating (Left), and after csignals when punishing (Right)

Figure 5 presents the cooperation rates after histories along which a subject has
consistently chosen ati = C but has observed either one bad signal in the previous
round or two consecutive bad signals in the two previous rounds. In other words,
the height of the three points on the graph in the left panel corresponds to the valuesof38
Pr ati = C | Cc, . . . , Cc ,Pr ati = C | Cc, . . . , Cc, Cd , andPr ati = C | Cc, . . . , Cc, Cd, Cd ,t−1where the history ht−1
such that (a1i , ω1i ) = · · · = (at−1i , ωi ) = (C, c) is abbreviatedi
as Cc, . . . , Cc, etc. As can be seen, the drop in the cooperation rates following asingle d signal is most conspicuous under perfect monitoring, suggesting the use
38

The figure only considers action choices in rounds three and above to allow for the observationof at least two signals.

23

of non-lenient strategies by the subjects. The rates under private monitoring are
similar to those under public monitoring but slightly lower. There is a statisticaldifference between perfect and either public or private, but there is no statisticaldifference between the last two.39
Although identification of histories that are relevant to forgiveness is less straightforward, we select the following histories as relevant. For subjects who started bycooperating: look at their first sequence of defection and compute the probability
that they cooperate when they observe one c signal or two consecutive c signals.
The height of the three points on the graph in the right panel of Figure 5 corresponds to the values ofPr ati = C | C∗, . . . , C∗, Dd, . . . , Dd ,Pr ati = C | C∗, . . . , C∗, Dd, . . . , Dd, Dc , andPr ati = C | C∗, . . . , C∗, Dd, . . . , Dd, Dc, Dc ,where C∗ implies either Cc or Cd. As can be seen, there is more forgiveness under
perfect than public following one cooperate signal, but less following two (theseare jointly statistically different from each other). Directionally, the comparison
between perfect and private is similar, but the rates following two c signals aremuch closer than perfect and public are. On the other hand, the difference between
public and private is not statistically significant. However, ranking the treatment
in terms of forgiveness on the basis of this exercise is difficult since the ranks varyfollowing one versus two cooperate signals.
For both leniency and forgiveness, the above analysis neglects most of the
choices in a supergame and hence is limited. We will return to the issue of leniency and forgiveness in the next section upon estimating strategies based on thesubjects’ action choices in all rounds of a supergame.
Observation 6 Conditional on some key histories, the levels of leniency are suchthatpublic > private  perfect.
The ordering in terms of the forgiveness levels is not as clear except that after asingle good signal in a punishment phase, a return to cooperation is most likely
under perfect monitoring.
39
This is established by regressing cooperation on a dummy if there was one d signal and interactedwith a dummy for the type of monitoring (as well as other controls).

24

7.5

Estimation of Strategies

We now turn to the direct estimation of the subjects’ strategies. Our analysis isbased on the Strategy Frequency Estimation Method (SFEM) developed in Dal Bó
& Fréchette (2011). SFEM has now been used in multiple papers to estimate thestrategies in repeated games, and its use is supported by Fudenberg et al. (2012)
who conduct Monte-Carlo simulations to evaluate its performance, and Dal Bó &Fréchette (2013) who find that the strategies identified as most popular by SFEM
are also the most popular strategies elicited from the subjects using an alternative
method.40 In essence, SFEM uses the maximum likelihood to estimate a mixture
over a given set of strategies.41 The parameters that are recovered represent theestimated fraction φk of strategy k in the set, and the variance in the distributionof the error term. Instead of reporting the parameter capturing the variance in the1 
error term, γ, we report the implied probability β ≡−1 that a cooperative
1+exp

γ̂

action would be taken when it is prescribed by a strategy. This gives an idea of
how well the model fits the data since β → 1 as γ → 0, and β → 12 (a coin toss) as
γ → ∞.
We consult previous studies that use SFEM in PD games to determine which
strategies to include. Specifically, we include all strategies that were found in a statistically significant proportion in any of the following papers: Dal Bó & Fréchette(2011); Fudenberg et al. (2012) and their re-analysis of Dal Bó & Fréchette (2011)and Dreber et al. (2008); Dal Bó & Fréchette (2013); Fréchette & Yüksel (2013);
and Embrey et al. (2013). The strategies included in our analysis are listed in Table
3 and are formally described by finite automata in Appendix A.2.
The three strategies in the top panel of Table 3 do not condition on the history:“always cooperate” (AllC), “always defect” (AllD), and a strategy that cooperates
in the first round and defects in all other rounds (CDDD). The second panel includes the well-known strategies of “grim trigger” (Grim) and “tit-for-tat” (TFT)
as well as their variants that either do not trigger a punishment after a single dor do not immediately return to cooperation following a single c: Grim2, Grim3,
TF2T, TF3T, 2TFT, and 2TF2T.42 Also in the second panel is the Sum2 strategy
that counts the numbers of good and bad signals: It has an internal counter that is40
Other papers using SFEM in different contexts include Vespa & Wilson (2015) on dynamic
games and Bigoni et al. (2015) on continuous-time games.
41
Intuitively, the method can be described as looking for the strategy from some given set that best
explains the observed choices of a subject in multiple supergames. It then looks for the frequency ofeach strategies in the entire sample. See Dal Bó & Fréchette (2011) for the details.
42
TFT in the imperfect monitoring environment starts by cooperating and then chooses C if andonly if ωi = c. In short, Grim-k is a variant of Grim that reverts to D after k consecutive d signals,
and m-TF-n-T is a variant of TFT that plays D in at least m consecutive rounds after n consecutive dsignals.

25

Lenient
AllC
AllD
CDDD
WSLS
Sum2
Grim
Grim2
Grim3
TFT
2TFT
TF2T
2TF2T
TF3T
STFT
SSum2

Forgiving

X
X

X
X
X

X
X
X
X
X
X
X

X
X
X
X

Complexity
1
1
2
2
4
2
3
4
2
3
3
4
4
2
4

Table 3: Properties of strategies

26initially set equal to zero. The counter is increased by 1 every time a good signal isobserved and the current value is below 2, and is decreased by 1 every time a badsignal is observed and the current value is above −2. The counter is unchanged inother cases. Sum2 plays C if the counter is ≥ 0 and D otherwise. Sum2 was first
explored in Embrey et al. (2013). The second panel also lists the “win-stay, loseshift” strategy (WSLS, sometimes referred to as “Pavlov” or “perfect tit-for-tat”)
that is known to have some desirable properties in environments with noise (Imhofet al. (2007)) but has not been found in statistically significant proportions in any
experiment we are aware of. Every strategy in the second panel yields a sequenceof (C, C)’s when matched against itself. On the other hand, the third panel lists thesuspicious versions of TFT and Sum2 (STFT and SSum2) that start by defecting,and yield a sequence of (D, D)’s when matched against itself. The check marks
in Table 3 show whether these strategies are lenient and/or forgiving.43 The lastcolumn of the table also shows the level of complexity of each strategy by the scale
1-4: It equals the number of states required when they are expressed as a finite
automaton.44
The results for perfect and public monitoring reproduce some results documented in the literature. First, under perfect monitoring, the majority of the data
can be accounted for by the three strategies: AllD, Grim, and TFT (Dal Bó &Fréchette (2014)). Second, lenient and forgiving strategies are more popular under imperfect monitoring than under perfect monitoring (Fudenberg et al. (2012)).
Third, despite its theoretical appeal, WSLS is not observed in any significant proportion in any treatment. Fourth, as in Embrey et al. (2013), Sum2 is observedin a statistically significant proportion. In other words, our results show that thefindings in the literature are robust with respect to the specifications of perfect andpublic monitoring such as randomly generated payoffs for perfect monitoring, introduction of noise into observation rather than into action choice under public
monitoring, and the cardinality and dimension of the signal space.
With respect to private monitoring, we first notice the prevalence of Sum2,
which was first documented in Embrey et al. (2013) in the public monitoring environment with a linearly ordered binary signal. Second, Grim and TFT are muchless popular than under perfect monitoring. In fact, TFT and all of its variants are
not very popular, and none of them is statistically significant individually (nor are
they jointly significant). Third, the lenient versions of Grim (Grim2 and Grim3) are
more popular than in any other treatment. Although Grim2 is not statistically significant on its own, its frequency is relatively high at 9% and Grim2 and Grim3 are
43

Classification of leniency and forgiveness is applied only to those strategies that have transitions
that depend on signals. Note that Sum2 and SSum2 are not lenient in all situations. For instance,
they will play D after a single d signal in the first round.
44
See Rubinstein (1986).

27

AllC
AllD
CDDD
WSLS
Sum2
Grim
Grim2
Grim3
TFT
2TFT
TF2T
2TF2T
TF3T
STFT
SSum2
Alpha
β
Cooperative
Noncooperative
Lenient
ForgivingComplexity = 1
Complexity = 2
Complexity = 3
Complexity = 4

Perfect

PublicPrivate0.024(0.032)0.314∗∗(0.062)0.000(0.004)0.022(0.045)0.000(0.007)0.117∗∗(0.050)0.046(0.034)0.023(0.038)0.176∗∗(0.042)0.000(0.000)0.108∗∗(0.042)0.079∗(0.045)0.076∗(0.042)0.015(0.029)0.000
0.471∗∗(0.040)
0.893
0.671
0.329
0.332
0.461
0.338
0.330
0.155
0.178

0.196∗∗(0.067)0.191∗∗(0.057)0.000(0.025)0.029(0.028)0.114∗(0.066)0.035(0.028)0.000(0.000)0.025(0.043)0.000(0.034)0.039(0.034)0.129∗∗(0.059)0.157∗∗(0.068)0.059(0.074)0.000(0.045)0.027∗
0.474∗∗(0.038)
0.892
0.783
0.217
0.484
0.526
0.387
0.064
0.168
0.381

0.032
(0.038)0.271∗∗(0.077)0.000(0.014)0.000(0.040)0.195∗∗(0.056)0.0138(0.040)0.090(0.056)
0.097∗∗(0.035)0.075(0.046)0.056(0.066)0.062(0.049)0.000(0.010)0.061(0.049)0.032(0.030)0.014
0.569∗∗(0.042)
0.853
0.683
0.317
0.505
0.450
0.304
0.121
0.208
0.367

*** Stat. sig. at the 1%, ** 5%, and * 10% levels.
Top panel are unconditional strategies, 2nd panel
are conditional and cooperative, and 3rd panel
are conditional and defective.
Bottom panel are total frequencies by feature.

Table 4: Estimates of proportion of each strategy
28

jointly statistically significant (p < 0.01). This is in sharp contrast with perfect andpublic monitoring, where neither of these two strategies is ever statistically significant. Fourth, going from perfect to private, strategies become more lenient, but notmuch more forgiving: 51% of strategies are lenient under private monitoring while
33% are lenient under perfect monitoring (p = 0.13). On the other hand, 15% are
non-lenient under private monitoring whereas 32% are non-lenient under perfectmonitoring (p < 0.1). In fact, the proportion of lenient strategies under private
monitoring (51%) is similar to that under public monitoring (48%) (p-value = 0.86
for the equality between public and private). As for forgiveness, the proportionof forgiving strategies is lower under private monitoring (45%) than under public
monitoring (53%) (insignificant at p = 0.52), and the proportion of unforgiving
strategies is higher under private monitoring (20%) than under public monitoring(6%) (again insignificant at p = 0.12). On the other hand, the forgiveness level
under private monitoring is similar to that under perfect monitoring.
This finding on forgiveness hence is at odds with the result of the reduced
form approach in Section 7.3 that the highest forgiveness level after a single csignal was under perfect monitoring. The discrepancy may come from the factthat the analysis in Section 7.3 is restricted to behavior after particular histories.
The conflicting findings are however partly reconciled by the observation that theestimated fraction of the TFT variants that return to cooperation immediately after
a single c signal (TFT, TF2T, and TF3T) is higher (0.36 in total) under perfectmonitoring than under public monitoring (0.19) or private monitoring (0.20).
In relation to leniency and forgiveness, we find that the strategies become
more complex when monitoring becomes imperfect whether it is public or private. Specifically, the estimated proportion of strategies which has just two statesin the automaton representation is higher under perfect monitoring (33%) than under either public (6%, p < 0.01) or private (12%, p < 0.05) monitoring. Likewise,
the estimated proportion of strategies with three or four states is one third under
perfect monitoring but slightly more than a half in both public and private monitoring (p < 0.05 for both comparisons). The average number of states equals 2.172,
2.544, and 2.638 for perfect, public, and private, respectively.
The most important differences across treatments can be gleaned by focusing
on the top three strategies in each treatment as listed in Table 5.
Notice that top three strategies represent more than 50% in proportion in all threetreatments. While the non-cooperative strategy AllD is always very popular, characteristics of cooperative strategies are markedly different in the three treatments.
Under perfect monitoring, both Grim and TFT are non-lenient. Under perfect monitoring, 2TF2T is both lenient and forgiving. Under private monitoring, both Sum2
and Grim3 are lenient, but the latter is non-forgiving. It is only under private monitoring that the intuitive strategy of Sum2 that counts the numbers of c’s and d’s is29Popularity
1 st
2nd
3rd

Monitoring
Perfect Public Private
AllD
TFT
Grim

AllC
AllD
2TF2T

AllD
Sum2
Grim3

Table 5: Top strategies by treatmentin the top three.
Observation 7 Strategies under public and private monitoring are more complex
than those under perfect monitoring. The increased complexity comes mainly fromthe lenient and forgiving variants of TFT under public monitoring, and from thelenient (but not forgiving) variants of Grim under private monitoring. Under bothpublic and private monitoring, Sum2, which counts the numbers of good and badsignals and is sometimes lenient and forgiving, is important.

8

Conclusion

While theory suggests the importance of a monitoring structure on the play of a repeated game, experimental work on the subject is still limited. This paper presents
one approach to the problem by comparing three major monitoring structures using
the same PD as a stage game.
Our findings from the perfect and public monitoring treatments serve as robustness checks of earlier results in the experimental literature as follows. First,
we find that cooperation is sustained even when payoffs are randomly determined
if the actions are perfectly monitored. This observation is in line with the findingof Rand et al. (2015) on an infinitely repeated PD that subjects for the most partignore random outcomes if they know their opponents’ intentions. Second, we observe cooperation under imperfect public monitoring which is specified differently
from that in prior experiments.45 Third, we confirm the key finding from Fudenberg et al. (2012) that strategies become more lenient and forgiving under public
monitoring than under perfect monitoring. We find this true under an alternative
specification of each monitoring structure as well as under an additional control on
the expected stage payoffs across the two monitoring treatments.
The primary focus of our analysis is on the comparison of private monitoringwith perfect and public monitoring. While theory suggests the difficulty of co45

See Dal Bó & Fréchette (2014) for a review.

30

operation under private monitoring, we observe that the subjects maintain almost
the same level of cooperation under private monitoring as under perfect and public monitoring. Even more surprisingly, the rates of coordination on either (C, C)or (D, D) is significantly higher than the hypothetical rates that would be obtained
when those actions are chosen independently at the observed rates.
We also find substantial difference in behavior under the three monitoring structures. Specifically, a reduced form approach based on choices after key histories,and strategy estimation both reveal that the behavior is more lenient under private
monitoring than under perfect monitoring. In both cases, the leniency level is similar under private and public monitoring. On the other hand, the comparison of
forgiveness levels is less clear. While strategy estimation suggests that similar forgiveness levels under perfect and private monitoring, and a higher level under public monitoring, the reduced form approach suggests the highest forgiveness level
under perfect monitoring.
In relation to leniency and forgiveness, we find that the strategies become more
complex when monitoring becomes imperfect whether it is public or private. In
particular, the average number of states in the finite automaton representation increases as we move from perfect to public, and from public to private. Underprivate monitoring, Sum2, which is complex and uses four states, is found to be
one of the top three strategies. It is interesting to note that as in the public monitoring treatment of Embrey et al. (2013), where Sum2 is first observed, our private
monitoring treatment has the feature that the signal is binary and can be interpreted
as either good or bad.
As in Matsushima & Toyama (2013), our finding regarding responsiveness is
not necessarily consistent with the play of a memory-one belief-free equilibrium,
which is widely used in the repeated game literature. Specifically, if subjects played
the memory-one belief-free equilibrium in all treatments, then the responsivenesslevel should be lowest under perfect monitoring and the same under public andprivate monitoring. We observe instead that the level under perfect monitoringis the highest. This implies at least that such an equilibrium is not played in alltreatments. Along with the results of our strategy estimation, we suspect that the
restriction to memory one is among other reasons for the observed deviation.
A full account of the behavior reported in this experiment would require thedevelopment of a new theory based on the combination of such elements as thecomplexity cost of strategies, preference for efficiency, and the importance of intentions: The fact that the estimated strategies are more complex in a more complex
environment suggests that the complexity of a strategy is perceived as a cost by the
subjects. In other words, simple strategies are preferred so long as they entail noefficiency loss. The Sum2 strategy is popular in the private monitoring treatment
perhaps because it is considered the simplest rule of thumb that works in the en31

vironment. Substantial rates of cooperation and coordination in every treatment
and leniency of strategies in the imperfect monitoring treatments both imply preference for efficiency. The higher responsiveness under perfect monitoring suggeststhat the subjects have a stronger incentive to react to the opponent’s action when
his intention is clearer. The reaction can also be used to discipline the opponent
when there is little or no noise. We view these as interesting insights to guide future
theory work on repeated games.

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A

Appendix

A.1

Cooperation under perfect monitoring in Dal Bó & Fréchette(2013).

In Dal Bó & Fréchette (2013), each session in the δ = 0.5 treatment has at least
19 supergames, while the three sessions in the δ = 0.9 treatment have 12, 18 and19 supergames. Given that there are at most 19 supergames in the current experiments, Figure 6 includes at most 19 supergames to make comparison easier. Dal
Bó & Fréchette (2013) specify the stage-payoffs as ui (D, C) = 50, ui (C, C) = 32,
ui (D, D) = 25, and ui (C, D) = 12, making the stage-game strategically equivalent13
to (2) for g = 25
7 − 1 ≈ 2.57 and ` = 7 ≈ 1.86.

1Round 1
δ=0.5

0

.2

Frequency of Cooperation
.4
.6
.8

δ=0.9

0

5

10
Supergame

15

20

Figure 6: Cooperation rates in Dal Bó & Fréchette (2013) by supergame

36

A.2

Strategies included in the estimation

Automaton
name in text

Diagram

AllC

-C
E

AllD

,D
E

Privatet1 = {ai = C, ω = (c, c)}t2 = ¬t1

t1 = {ai = C, ωi = c}t2 = ¬t1



Dr

C
E

CDDD

Perfect and Public

t2
t1



Dr

-C
E

Grim

t2
t1

TFT



t2

t1 = {ωi = c}t2 = ¬t1



t1

t1 = {ωi = c}t2 = ¬t1

Dr

-C
EQ
t1
t2
t1

Dr

-C
EQ

WSLS

t2
t2
t1

STFT



t2

t1 = {ωi = c}t2 = ¬t1

DY r

-C
Q
t1

Table 6: Unconditional and Two-States Automata

37

Automatonname in text

Diagram
t2
t1



-C
EQ

Grim2

Perfect and Public

Privatet1 = {ai = C, ω = (c, c)}t2 = ¬t1

t1 = {ai = C, ωi = c}t2 = ¬t1

t1 = {ai = C, ω = (c, c)}t2 = ¬t1

t1 = {ai = C, ωi = c}t2 = ¬t1

t2



C

Dr

t2

t2

t1
t2
t1



-C
E QS

Grim3



C
t1

t1
t2

t2

t1



-C
E QS

TF2T

t2

t2



-C
E QU



CQ

t2



-C
ES

DQ r

t2

t2

t1



t2
t1
t2



D

t1

t1





DQ r

C

4 FC h

t1

(

Cq

t2

FC h
t1 t2

*Dh

t1

(

Dr

t1

(

t2
t2
t1

t1 = {ωi = c}t2 = ¬t1

t2

t2

Dh

t1 = {ωi = c}t2 = ¬t1

t2

t1

t2

t1 = {ωi = c}t2 = ¬t1

D



-C
E QU
t1

SSum2

t1 = {ωi = c}t2 = ¬t1

t1
t2

t1

Sum2

Dr

t1

t1

2TF2T

t2



C

t1

2TFT

t1 = {ωi = c}t2 = ¬t1

t1

t2
t1

t2



Dr

C
t1

TF3T



Dr

C

(

t1

Cq
t2

Dr

38
Table 7: Automata with More Than Two States

t1 = {ωi = c}t2 = ¬t1