2016-12
Finitely Repeated Games with Automatic andOptional Monitoring

Yasuyuki Miyahara Tadashi Sekiguchi

Finitely Repeated Games with Automatic andOptional Monitoring⋆
Yasuyuki Miyaharaa ∗Kobe University

Tadashi Sekiguchib
Kyoto University

May 10, 2016
Abstract
We study a new class of finitely repeated games with optional monitoring,where each player automatically receives complete information about the otherplayers’ actions with some exogenously given probability. Only when theautomatic information did not arrive, the player privately decides whether toexercise a costless monitoring option or not. We show that a weak decrease in
the vector of the players’ probabilities of automatic monitoring is a necessary andsuﬃcient condition for any repeated game with automatic and optional monitoring
to have a weakly greater sequential equilibrium payoﬀ vector set. This result
considerably strengthens our earlier result, which only compares purely optional
monitoring and the standard model of purely automatic monitoring. We also
provide examples where existing folk theorems hold under any automatic andoptional monitoring structure but not under the standard model.
JEL Classification: C72; C73
Keywords: Repeated games; Monitoring option; Imperfect monitoring; Folk
theorem

1.

Introduction

This paper studies repeated games where monitoring is part of the players’ decision
making. Particularly, we present a new class of repeated games with optional⋆

We thank Masatoshi Tsumagari for a stimulating conversation which originated this research.
We are grateful to audience at the MEA meeting, Hitotsubashi University, Decentralization
Conference in Japan, UECE Lisbon Meetings, and SING11-GTM2015. We gratefully acknowledge
financial supports from the Grants-in-Aid for Scientific Research (Miyahara, 24530199; Sekiguchi,
23000001, 26245024, 26380238). This research was completed when Miyahara stayed at Yale
University. Miyahara gratefully acknowledges the hospitality of Yale University.
aGraduate School of Business Administration, Kobe University, 2-1 Rokkodai-cho, Nada-ku, Kobe
657-8501, Japan∗
Corresponding author.
E-mail addresses: miyahara@kobe-u.ac.jp (Y. Miyahara), sekiguchi@kier.kyoto-u.ac.jp (T.
Sekiguchi)bInstitute of Economic Research, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto
606-8501, Japan1

monitoring, where each player can costlessly decide whether to monitor the otherplayers’ actions or not. This paper introduces a possibility that each player may
automatically learn the others’ actions with some exogenously given probability, andthe players’ monitoring decisions are relevant only when the automatic information
did not arrive.
More concretely, we set up a model of finitely repeated games with the following
structure. In each period, after the players have chosen their stage-game actions, eachplayer automatically receives complete information about the other players’ actions
with some exogenous probability which does not depend on the actions. We assume
that arrivals of the automatic information are independent across the players andover time. If the automatic information did not arrive, the player privately decides
whether to exercise his monitoring option or not. If he exercises the option, he learns
the others’ actions without any noise. Otherwise, he does not get any information
about them.1 It is costless to monitor the other players, and the monitoring decision
is completely unobservable. Namely, each player receives no signal as to whether any
other player learnt his action either by automatic monitoring or his own monitoring.
Our earlier paper (Miyahara and Sekiguchi [9]) studies the case of purely optional
monitoring, where the automatic information never arrives.
We have three results. The first two results examine how the sequential
equilibrium payoﬀ vector set of the repeated games with automatic and optional
monitoring depends on the vector of the players’ automatic monitoring probabilities,
which we simply call the monitoring vector. We first show that for any stage game,
any number of repetition, and any discount factor, the sequential equilibrium payoﬀ
vector set of the corresponding repeated game weakly decreases (in the sense of set
inclusion) if the monitoring vector weakly increases. In other words, any sequential
equilibrium payoﬀ vector under a given monitoring vector continues to be a sequential
equilibrium payoﬀ vector if the monitoring vector decreases to any direction.
Second, and more interestingly, we show that a converse of the above result is also
true. Fix two n-dimensional monitoring vectors λ and λ′ such that λ ≥ λ′ does nothold. Then, an n-player game exists such that the two-period repeated game underλ′ has a strictly smaller sequential equilibrium payoﬀ vector set than the two-period
repeated game under λ if the players are patient enough. Together with the first
result, we see that a weak decrease in the monitoring vector is a necessary andsuﬃcient condition for any repeated game with automatic and optional monitoring
to have a weakly greater sequential equilibrium payoﬀ vector set. Those results
considerably strengthen the results of [9], which just compares the standard modelof purely automatic monitoring and the case of purely optional monitoring.
Our third result is as to when a folk theorem (like Benoı̂t and Krishna [1],Smith [10], and Gossner [4]) holds under any monitoring vector. It is easy to verify
that our first result extends when we compare purely automatic monitoring and our
automatic and optional monitoring. Therefore, if an existing folk theorem applies
to some stage game under purely automatic monitoring, the same theorem applies
1

This assumption denies a possibility that the player learns the others’ actions from his stage
payoﬀ. To this end, we will assume that the players collect their stage payoﬀs in total, at the endof the repeated game.

2

to the same stage game under any monitoring vector. Our third result proves thatits converse is not true. Namely, we show that for any n, an n-player stage gameexists such that (i) it has a unique Nash equilibrium, (ii) the full dimensionality
condition is satisfied, and (iii) for any monitoring vector, there exists T such that theundiscounted T -period repeated game with that monitoring vector has a sequential
equilibrium which gives each player a diﬀerent average payoﬀ from the stage-gameequilibrium. The second and third conditions allow us to apply existing folk theorems
for any monitoring vector. However, the first condition denies any folk theorem underpurely automatic monitoring. Hence, this result highlights a fundamental diﬀerenceof our model from the standard model.
One interpretation of our automatic and optional monitoring structure is players’
overlooking. Namely, precise information about their actions is released (for example,
sent by emails), and an absentminded player overlooks it. However, in case heoverlooked the information, he has an opportunity to retrieve it (for example,
by checking his email box). In this interpretation, the probability of automatic
monitoring is the probability that the player is not absentminded.
Another interpretation is failure in avoiding information. As the above argument
suggests, the players sometimes benefit from having a smaller monitoring probability,
because it may create a more eﬃcient equilibrium. Then they may want to install adevice which prevents arrivals of the automatic information.2 If the device is subject
to random malfunction, however, then the probability of malfunction corresponds tothe probability of automatic monitoring.
A key insight into our results is that since optional monitoring is costless andcompletely unobservable to the others, a player never hurts from having a smaller
probability of automatic monitoring. The own monitoring simply compensates forlack of automatic information, because it is impossible to detect it and to inflictpunishment on the monitor. Rather, the smaller probability of automatic monitoring
expands their strategic flexibility. This is true in [9], but it only considers an extreme
case where all players’ probability of automatic monitoring changes from one to zero.
We rather reveal that even a change that decreases only one player’s automatic
monitoring probability in a slightest way is, in some cases, suﬃcient to create newstrategic possibilities.
Our second and third results are built on an example with a unique stage-gameequilibrium. Namely, this is an example of a finitely repeated game with a nontrivial
equilibrium despite that the stage game has a unique equilibrium, which never arises
under the standard model.3
It is worthwhile to point out that the nontrivial equilibrium has a feature thata player randomizes over his monitoring decisions in case the automatic information
did not arrive. Since the monitoring decision is private, a potential deviator does
not know whether his deviation will be detected or not. The player who did notobserve the others cannot respond to the deviation, and sequential rationality only
limits behavior of the deviator and the other players knowing the deviation. This2

At the same time, the device must maintain their ability to monitor the others if they wish.
However, this feature is not novel and is known in [9]. Further, other repeated games withimperfect monitoring, such as Kandori [5] and Mailath et al. [7], exhibit this type of results. See
also a recent paper by Sugaya and Wolitzky [11].
33

limited requirement of sequential rationality, caused by lack of common knowledgeof the deviation, creates a credible punishment which works even if the stage gamehas a unique equilibrium. Note that a greater probability of automatic monitoringis burden for this construction, because it weakens the impact of the players who do
not notice the deviation.
A main body of literature on repeated games with endogenous monitoring rather
assumes that it is costly to monitor the others (for instance, Ben-Porath andKahneman [2], Kandori and Obara [6], Miyagawa et al. [8], and Flesch and Perea [3]).
In those papers, a central question is provision of incentives to monitor. Clearly,
adding a possibility of automatic monitoring to those frameworks has quite diﬀerent
eﬀects from ours, because the automatic information mitigates the incentive problem
about costly monitoring. It would be interesting to investigate those eﬀects morethoroughly.
The rest of this paper is organized as follows. Section 2 introduces the model.
Section 3 proves that a weak decrease in the monitoring vector is a necessary andsuﬃcient condition for any repeated game with automatic and optional monitoring
to have a weakly greater sequential equilibrium payoﬀ vector set. Section 4 provides
examples where existing folk theorems hold under any automatic and optional
monitoring structure but not under the standard model. We leave some proofs to theappendices.

2.

Model

Let G be a finite, n-player strategic form game with n ≥ 2, and let Ai be the setof player i’s pure actions. Player i’s payoﬀ function is given by ui : A → R, whereA ≡ A1 × · · · × An denotes the set of pure action profiles. Each player can choose amixed action, and ∆Ai denotes the set of player i’s mixed actions.
After the players have chosen their actions, each player i automatically receives
complete information about the other players’ actions with probability λi ∈ (0, 1). We
call λi player i’s automatic monitoring probability. We assume that each λi does notdepend on the actions and that the arrivals of automatic information are independent
across the players. Let λ = (λ1 , . . . , λn ) be the vector of automatic monitoring
probabilities, called the monitoring vector hereafter. If the automatic information
did not arrive, each player privately decides whether to monitor his opponents or
not.4 Monitoring the others is costless.
Fix δ ∈ (0, 1] and an integer T ≥ 2. Let Gδ (λ, T ) be the δ-discounted
T -period repeated game with automatic and optional monitoring where G is playedin periods t = 1, . . . , T . In each period t ≥ 1, each player i chooses an action ai ∈ Ai
simultaneously. Then, player i decides whether to monitor all the other players
depending on his choice of action, in case the automatic information did not arrive.
Each player can choose randomly whether to monitor them or not. We assume thatthe monitoring decision is not observable to the other players. Therefore, our model4

We assume that the monitoring decision is binary in order to simplify the notations. It is easy
to extend the analysis to the case where players can monitor any subset of the players.

4belongs to one with private monitoring.5
We also assume that if a player does not monitor the other players, then hereceives no information about their actions. This amounts to assuming that theplayers receive all stage payoﬀs at the end of the repeated game, because the stage
payoﬀs in general provide information about the actions. The players can monitor
the actions in a period only at the end of that period; it is impossible to acquire
information of any past period.
The information player i obtains in each period t is his action and information
about the other players’ actions in that period. We define Ii = [A × {0, 1}] ∪ Ai
as the set of information player i obtains in one period. Here, (i) (a, 0) ∈ Ii meansthat player i chose ai and then automatically learnt that the other players’ actions
were a−i , (ii) (a, 1) ∈ Ii means that player i chose ai , did not receive the automatic
information, and then found the other players playing a−i by his own monitoring, and(iii) ai ∈ Ii means that player i chose ai , did not receive the automatic information,and did not monitor the other players.
Player i’s history at period t ≥ 2 consists of all his past information he obtains
up to the end of period t − 1. For t ≥ 2, the set of all histories for player i at period t
is Hit = (Ii )t−1 . Let Hi1 be an arbitrary singleton set. The set of player i’s histories
at all periods is
Hi =

T
∪

Hit .

t=1

A strategy of player i is denoted by σi = (σia , σim ). Here, σia prescribes player i’s
mixed action at each history at each period, namely, σia : Hi → ∆Ai . Then σimprescribes a probability that player i monitors all the other players when automatic
information did not arrive, given any history at any period and any stage-game action
chosen in that period, namely, σim : Hi × Ai → [0, 1]. We call an element of Hi × Ai
a history-action pair.
Given a strategy profile σ = (σ1 , . . . , σn ), player i’s payoﬀ in Gδ (λ, T ) is
[ T
]
∑
(
)
1−δ
E
δ t−1 ui a(t) ,1 − δT
t=1
where a(t) is the action profile in period t, and the expectation is taken with respectto σ and arrivals of automatic information.
The solution concept of this paper is sequential equilibrium adapted to our finitely
repeated game. A system of beliefs is a function which maps each history hti toa probability distribution of the other players’ history profiles (htj )j̸=i and maps
each history-action pair (hti , ai ) to a probability distribution of the other players’
history-action pair profiles (htj , aj )j̸=i . A strategy profile is completely mixed if at
any history, any stage-game action is selected with positive probability and at any
history-action pair, any monitoring decision is selected with positive probability. A
tremble of a strategy profile is defined as a sequence of completely mixed strategy
profiles converging to the strategy profile. Given a strategy profile σ, a system
5[9] studies the case of λ = (0, . . . , 0).

5

of beliefs is consistent if there exists a tremble of σ such that the corresponding
sequence of the system of beliefs, obtained from Bayes’ rule, converges to it.6 A
strategy profile σ is a sequential equilibrium, if there exists a consistent system ofbeliefs ψ such that (a) at any history of any player i, his continuation strategy is
optimal given σ−i and the belief about the other players’ histories specified by ψ, and(b) at any history-action pair of any player i, his continuation strategy is optimal
given σ−i and the belief about the other players’ history-action pairs specified by ψ.

3.

Monotonicity in the Monitoring Vector

This section examines how the sequential equilibrium payoﬀ vector set of the repeatedgames with automatic and optional monitoring depends on the monitoring vector.
Theorem 1. Fix n ≥ 2 and two n-dimensional monitoring vectors λ and λ. Thenthe following two conditions are equivalent.
(A) λ ≥ λ.
(B) For any n-player stage game G, any δ ∈ (0, 1], and any integer T ≥ 2, any
sequential equilibrium payoﬀ vector of Gδ (λ, T ) is a sequential equilibrium payoﬀ
vector of Gδ (λ, T ).
Theorem 1 implies that for any G, any T , and any δ, the set of sequential
equilibrium payoﬀ vectors of Gδ (λ, T ) is weakly decreasing in the monitoring vector,
in the sense of set inclusion. The theorem also implies that for any change in themonitoring vector other than a weak decrease, a discounted finitely repeated gameexists such that a payoﬀ vector attained by a sequential equilibrium before the change
cannot be sustained by a sequential equilibrium after the change. To summarize, aweak decrease in the monitoring vector is a necessary and suﬃcient condition for
any repeated game with automatic and optional monitoring to have a weakly greater
sequential equilibrium payoﬀ vector set.
Appendix A proves that (A) implies (B). The intuition behind this part is
simple. Since monitoring is costless and private, the players can always compensate asmaller probability of automatic monitoring by increasing the probability of his own
monitoring, without aﬀecting their payoﬀs and beliefs. Therefore, we can modify any
sequential equilibrium of Gδ (λ, T ), so that the modified strategy profile of Gδ (λ, T )has exactly the same play as the original equilibrium.
The rest of this section verifies that (B) implies (A). Fix n ≥ 2 and two
n-dimensional monitoring vectors, λ and λ, such that λ1 < λ1 . Note that, without
loss of generality, this is a denial of λ ≥ λ. It suﬃces to verify existence of ann-player stage game G, δ ∈ (0, 1], and T ≥ 2 such that Gδ (λ, T ) has a strictly smaller
sequential equilibrium payoﬀ vector set than Gδ (λ, T ).7 The next result, however, is
6

For a consistent system of beliefs and for any hti and ai , there is a close connection between the
beliefs at hti and at (hti , ai ). Namely, the probability of any history-action pair profile (htj , aj )j̸=i at(hti , ai ) is the product of the probability of (htj )j̸=i at hti and the probability of (aj )j̸=i under theother players’ mixed actions at (htj )j̸=i .
7
Here we reverse the roles of λ and λ in the statement of Theorem 1.

6somewhat stronger in the following senses. First, we can set T = 2, so that a very
short horizon is suﬃcient to create a diﬀerence. Second, we can arbitrarily fix δ, sothat patience is not indispensable for creating a diﬀerence. For that matter, the samestage game which works for the fixed δ also works for any greater discount factor.
Third, the choice of G depends only on λ1 and λ1 , so that the values of λi and λi for
any i ≥ 2 are irrelevant. For that matter, for any µ ∈ [λ1 , λ1 ), we can choose G so′that it works not only for λ and λ, but also for any two monitoring vectors λ′ and λ′such that λ′1 ≤ µ < λ1 .
Proposition 1. Fix n ≥ 2, µ ∈ (0, 1), and δ ∈ (0, 1]. Then, an n-player strategic
form game G exists such that for any λ and λ such that λ1 ≤ µ < λ1 , Gδ (λ, 2)
has a unique sequential equilibrium payoﬀ vector for any δ, and Gδ (λ, 2) with anyδ ≥ δ has a sequential equilibrium whose payoﬀ vector is diﬀerent from the sequential
equilibrium payoﬀ vector of Gδ (λ, 2).
Proof. Fix n ≥ 2, µ ∈ (0, 1), and δ ∈ (0, 1]. Define G so that A1 = {U, M, D},
A2 = {L, C, R}, and if n ≥ 3, Ai = {bi , ci } for any i ≥ 3. u1 and u2 depend only on(a1 , a2 ), and are represented by the following payoﬀ matrix.
L

C

R

U

4, 4

0, 3

µ
4, 3 − 1−µδM0, 3 − 1−µµ6, 30, 3 + δ

D

3, 6

3, 0

3, 6

Figure 1: Payoﬀ matrix for players 1 and 2
For i ≥ 3 (if any),




3
ui (a) = 20if (a1 , a2 ) = (M, C) and ai = bi ,if (a1 , a2 ) ̸= (M, C) and ai = bi ,(1)

if ai = ci .

First, we show that G has a unique Nash equilibrium (U, L, b3 , . . . , bn ). It
is suﬃcient to show that C is not played with positive probability in any
Nash equilibrium, because we have a unique outcome (U, L, b3 , . . . , bn ) by iterated
elimination of strictly dominated actions in the reduced game obtained after
eliminating C from the set of actions of player 2.
Suppose that player 1 plays U with probability x, M with probability y, and Dwith probability 1 − x − y in a Nash equilibrium of G. A necessary and suﬃcientcondition for C to be a best response of player 2 is
(1 − µ)y(1 − µ)y 6(1 − µ)+(1 − x − y) ≤ x ≤− 6(1 − x − y).
µδµµThis condition implies 1−x−y = 0 and x = (1−µ)y/µ, which in turn imply x = 1−µand y = µ. Since x > 0 and y > 0, player 1 must be indiﬀerent between U and M ,7and that holds only if player 2 plays C with probability 2/5. Then, D is a uniquebest response of player 1. Therefore, C is not played with positive probability in any
Nash equilibrium. Hence, (U, L, b3 , . . . , bn ) is a unique Nash equilibrium of G.
Fix λ and λ such that λ1 ≤ µ < λ1 . In what follows, we show that Gδ (λ, 2) withany δ ≥ δ has a sequential equilibrium payoﬀ vector which is diﬀerent from the unique
Nash equilibrium payoﬀ vector of G, and Gδ (λ, 2) with any δ ∈ (0, 1] has a uniquesequential equilibrium payoﬀ vector, which equals the unique Nash equilibrium payoﬀ
vector of G.
First, fix δ ≥ δ, and let σ̂ be the following strategy profile of Gδ (λ, 2).
• In period 1, player 1 plays M . Then if the automatic information did not arrive,
he monitors the other players with probability (µ − λ1 )/(1 − λ1 ), and does notmonitor them with probability (1 − µ)/(1 − λ1 ) irrespective of his action. In
period 2, player 1 plays M and monitors the other players irrespective of his
action if he found player 2 not playing C in period 1. Otherwise, player 1 plays
U and monitors the other players irrespective of his action.
• In period 1, player 2 plays C, and monitors the other players irrespective of his
action. In period 2, player 2 plays L and monitors the other players regardless of
his action if he played C in period 1. Otherwise, player 2 plays C and monitors
the other players regardless of his action.
• Player i ≥ 3 plays bi and monitors the other players regardless of his action at
any history.
The play under σ̂ is (M, C, b3 , . . . , bn ) in period 1 and (U, L, b3 , . . . , bn ) in period 2,
and the average discounted payoﬀ vector is()
6 + 4δ 3 + 4δ 3 + 2δ
3 + 2δ
,,,...,.
1+δ 1+δ 1+δ
1+δ
It is diﬀerent from the Nash equilibrium payoﬀ vector of the stage game. It remains
to show that σ̂ is a sequential equilibrium. Fix any consistent system of beliefs givenσ̂. First, we consider player 1.
• At any history of player 1 at period 2 such that player 1 found player 2 notplaying C in period 1, player 1 believes that player 2 plays C with probability
one in period 2. Hence, it is optimal for player 1 to play M with any monitoring
decision in period 2.
• At any history of player 1 at period 2 such that player 1 did not find a deviationof player 2 in period 1, player 1 believes that player 2 did not deviate and plays
L with probability one in period 2. Hence, it is optimal for player 1 to play U
with any monitoring decision in period 2.
• In period 1, player 1 plays a short-run best response, and his action does not
aﬀect future play. Further, he has nothing to learn from the others’ actions,because their strategies are pure. Hence, it is optimal for player 1 to play M
and randomize between monitoring and not monitoring in the way specified byσ̂ in period 1.
8Next, we consider player 2.
• At any history of player 2 at period 2 such that he played C in period 1, player 2
believes that player 1 plays U with probability one. Hence, it is optimal forplayer 2 to play L with any monitoring decision in period 2.
• At any history of player 2 at period 2 such that he did not play C in period 1,
player 2 believes that player 1 plays U and M with probabilities 1 − µ and µrespectively. This is because consistency requires that, whether player 2 foundplayer 1’s deviation or not, he believes that player 1 finds player 2’s deviation(and therefore plays M in period 2) with probability
λ1 + (1 − λ1 ) ·µ − λ1= µ.
1 − λ1Hence, it is optimal for player 2 to play C with any monitoring decision inperiod 2, and his stage payoﬀ is 3.
• When player 2 follows σ̂2 , his average discounted payoﬀ is (3 + 4δ)/(1 + δ). If
he does not play C in period 1, then his stage payoﬀ is at most 3 + δ. We haveseen that this deviation leads to a continuation play where his stage payoﬀ is 3
in period 2. Therefore, his average discounted payoﬀ when he does not play Cin period 1 is at most (3 + δ + 3δ)/(1 + δ). This implies that conforming to σ̂2is optimal since δ ≥ δ.
Finally, for any player i ≥ 3, his play does not aﬀect future play at all. Hence, it is
optimal to always play a static best response bi , as is prescribed by σ̂i . Therefore, σ̂
is a sequential equilibrium.
Next, fix δ ∈ (0, 1]. We show that Gδ (λ, 2) has a unique sequential equilibrium
payoﬀ vector. Let σ be a sequential equilibrium in Gδ (λ, 2).
Claim 1. For any i ≥ 3, σi prescribes to play bi at any history.
At any history of player i ≥ 3 at period 2, bi is uniquely optimal. At period 1,choosing ci instead of bi causes a loss of 2 or more in period 1, and the gain in period 2
is at most 1, because he can guarantee 2 by playing bi . Therefore, playing bi is always
optimal.
Let Â2 ⊆ A2 be the set of player 2’s actions played with positive probability inperiod 1 under σ. Also, let Ĥ22 be the set of player 2’s histories at period 2 suchthat he played a2 ∈ Â2 , monitored the other players by either automatic or own
monitoring, and found any player i ≥ 3 (if any) playing bi in period 1. Note that
player 2 may have observed a deviation of player 1 at h22 ∈ Ĥ22 .
Claim 2. At any history h22 ∈ Ĥ22 , player 2 does not play C with positive probability.
Fix h22 ∈ Ĥ22 , and let a1 and a2 be the actions of players 1 and 2 in period 1 under
h22 . Define a = (a1 , a2 , b3 , . . . , bn ). Then player 2 believes that player 1’s history at
period 2 is either (a, 0), (a, 1), or a1 .

9Suppose, on the contrary to the claim, that σ2 prescribes to play C with positive
probability at h22 . We have seen before that C is optimal only if player 2 believes that
player 1 plays U with probability 1−µ and M with probability µ. For player 1, U and
M are not simultaneously optimal at any history at period 2 (player 1 is indiﬀerent
between U and M only when player 2 chooses C with probability 2/5, but then D is
uniquely optimal). Since λ1 > 0, player 2 believes that player 1 is at h21 = (a, 0) withpositive probability, and also believes that player 1 at h21 = (a, 0) plays a pure action
other than D, which we denote by â1 .
Player 1 has the same belief about the other players’ actions at (a, 0) and (a, 1).
Thus, â1 is optimal at both (a, 0) and (a, 1). Consequently, in order to believe thatboth U and M are played with positive probability, player 2 must believe that player 1
is at h21 = a1 with positive probability and player 1 at h21 = a1 plays the pure actionã1 ∈ {U, M } \ {â1 }.
It follows from consistency that player 1 who chose a1 and did not receive
automatic information in period 1 does not monitor the other players with positive
probability. Hence, sequential rationality implies that player 1 who chose a1 anddid not receive automatic information finds it optimal not to monitor the others inperiod 1 and then play ã1 in period 2. Since a2 ∈ Â2 , player 1 at the time of his
monitoring decision believes that he reaches h21 = (a, 1) with positive probability if
he monitors the others. As a result, ã1 must be optimal at h21 = (a, 1), which inturn implies that ã1 must be optimal at h21 = (a, 0). This is a contradiction, which
establishes the claim.
Claim 3. Fix a2 ∈ Â2 , and suppose that σ2 prescribes not to monitor the other playerswith positive probability if he played a2 and did not receive automatic informationin period 1. Then at the history h22 = a2 , player 2 does not play C with positive
probability.
On the contrary, suppose that σ2 prescribes to play C with positive probability
at h22 = a2 . From sequential rationality, it must be optimal not to monitor the othersin period 1 and then to play C in period 2, if he played a2 and did not receive
the automatic information in period 1. This implies that it must be also optimal
to monitor the other players in period 1 and then to play always C in period 2, if
he played a2 and did not receive the automatic information in period 1. However,player 2 believes that he will surely reach a history in Ĥ22 , and we have seen in the
proof of Claim 2 that C is not optimal at any history in Ĥ22 . Therefore, monitoring theothers and then playing an optimal action for each realized history improve player 2’s
payoﬀ. This is a contradiction.
Claim 4. At any history of player 1 at period 2 such that he did not find any player’s
deviation in period 1, σ1 prescribes U with probability one.
At any such history, consistency requires player 1 to believe that any player did
not deviate in period 1. From Claims 2 and 3, player 2 never plays C if he did notdeviate in period 1 and did not find a deviation by player i ≥ 3. When C is notplayed with positive probability, U is uniquely optimal.
Claim 5. At any history of player 2 at period 2 such that he monitored the otherplayers in period 1, player 2 does not play C with positive probability.
10Let a ∈ A be the combination of player 2’s action and his observation in period 1,
given the history. Then player 2 believes that player 1’s history is either (a, 1), (a, 0),
or a1 . From Claim 4, player 1 plays U with probability one at the history a1 . Hence,
for C to be optimal, σ1 must prescribe M with positive probability at either (a, 1)or (a, 0). Since player 1’s belief about the others’ actions is the same at the two
histories, sequential rationality requires that M is optimal at both (a, 1) and (a, 0).
Therefore, σ1 does not prescribe U with positive probability at the two histories,
because U and M cannot be simultaneously optimal. Since player 1 receives theautomatic information and reaches the history (a, 0) with probability λ1 > µ, theprobability with which player 2 believes that player 1 plays U is less than 1 − µ.
Given the belief, C is not optimal.
Claim 6. Fix a2 ∈ A2 , and suppose that σ2 prescribes not to monitor the other playerswith positive probability if he played a2 and did not receive automatic informationin period 1. Then at the history h22 = a2 , player 2 does not play C with positive
probability.
On the contrary, suppose that σ2 prescribes to play C with positive probability
at h22 = a2 . From sequential rationality, it must be optimal not to monitor the othersin period 1 and then to play C in period 2, if he played a2 and did not receive theautomatic information in period 1. This implies that it must be also optimal tomonitor the other players in period 1 and then to play always C in period 2, if heplayed a2 and did not receive the automatic information in period 1. However, wehave seen in the proof of Claim 5 that C is not optimal at any history at period 2
where he played a2 and monitored the others in period 1. Therefore, monitoring
the others and then playing an optimal action for each realized history at period 2
improve player 2’s payoﬀ. This is a contradiction.
Claim 7. At any history of player 1 at period 2, σ1 prescribes U with probability one.
Consistency requires player 1 to believe that player 2 conforms to his monitoring
decision given his (possibly deviant) action. Hence, from Claims 5 and 6, player 1
believes that player 2 does not play C with positive probability in period 2. Given
the belief, U is uniquely optimal.
Claim 8. At any history of player 2 at period 2, σ2 prescribes L with probability one.
By Claim 7, player 1 plays U at any history at period 2. Thus, L is uniquely
optimal at any history at period 2.
From Claims 1 to 8, the players follow (U, L, b3 , . . . , bn ) at any history at period 2.
Given that, play in the first period must form a Nash equilibrium of G. Therefore,
the players play (U, L, b3 , . . . , bn ) in period 1. The payoﬀ vector of this sequential
equilibrium is (4, 4, 2, . . . , 2), which completes the proof. Q.E.D.
We have two remarks on Theorem 1. First, although we assume that each player’s
automatic monitoring probability is neither 0 nor 1, the results extend to the casewhere λi = 0 or λi = 1 for some i. This extension is straightforward, except fornotational cumbersomeness. If λi = 0, player i never receives automatic information.
Thus, we should redefine Ii , the set of information player i obtains in one period, as
11

Ii = A ∪ Ai . If λi = 1, the automatic information surely arrives. Thus, we should
redefine Ii as Ii = A. With these modifications, we can redefine each player’s histories
and strategies and study the repeated games accordingly. Then Theorem 1 continues
to hold even if λ and λ contain 0 and/or 1. We omit a proof, because the intuition
should be clear.
Second, we see from the proof of Proposition 1 that for any n-dimensional
monitoring vector λ, an n-player stage game G exists such that G has a uniqueequilibrium but Gδ (λ, 2) has multiple sequential equilibrium payoﬀ vectors if δ islarge enough. This further implies that for all large δ, Gδ (λ, T ) with any T ≥ 3
has multiple sequential equilibrium payoﬀ vectors.8 In the standard model of purely
automatic monitoring, if the stage game has a unique equilibrium payoﬀ vector,
any finitely repeated game has a unique subgame perfect equilibrium payoﬀ vector.
Therefore, our results imply that this well-known negative result does not extend toany automatic and optional monitoring environment.9

4.

Folk Theorem

In this section, we compare automatic and optional monitoring with purely automatic
monitoring in view of the folk theorem (Benoı̂t and Krishna [1], Smith [10], andGossner [4]).
As the remark at the end of the previous section indicates, if an existing folk
theorem applies to a given stage game under purely automatic monitoring, the same
folk theorem applies to the same stage game under any monitoring vector. The
following result, which is proved in Appendix B, shows that its converse does not
hold.
Theorem 2. For any n ≥ 2, there exists an n-player strategic form game G suchthat(i) G has a unique Nash equilibrium,(ii) the full dimensionality condition is satisfied, and(iii) for any monitoring vector λ, there exists T such that G1 (λ, T ) has a sequential
equilibrium which gives each player a diﬀerent payoﬀ from the Nash equilibriumof G.
The conditions (ii) and (iii) allow us to apply any existing folk theorem under
any monitoring vector.10 In contrast, the condition (i) implies that no folk theorem
holds under purely automatic monitoring. Therefore, the suﬃcient condition for a
For any T ≥ 3, playing a static equilibrium in all periods 1, . . . , T − 2 and then playing a
nontrivial sequential equilibrium of Gδ (λ, 2) in the last two periods form a sequential equilibrium ofGδ (λ, T ).
9[9] points out this failure of the negative result under purely optional monitoring.
10
Note that the condition (iii) limits attention to no discounting. This restriction is not essential,
and we impose it just because some folk theorems assume no discounting presumably for simplicity(e.g., Gossner [4]).
812folk theorem to hold under any automatic and optional monitoring structure is quite
diﬀerent from that in the standard model.11 Note that an arbitrarily small departurefrom purely automatic monitoring is suﬃcient for a folk theorem. Theorem 2 thus
highlights a fundamental diﬀerence between the standard model and ours.
We have two more remarks. First, the stage game G constructed in the proof of
Proposition 1 satisfies the conditions in Theorem 2. Proposition 1 implies existenceof λ such that Gδ (λ, 2) has a unique sequential equilibrium payoﬀ vector for any
δ. Further, Theorem 2 implies existence of T such that G1 (λ, T ) has a nontrivial
equilibrium. These results reveal another diﬀerence from the standard model. Under
purely automatic monitoring, if T exists such that the T -period repeated gamehas a unique subgame perfect equilibrium payoﬀ vector under any discount factor,
any finitely repeated game with any discount factor has a unique subgame perfectequilibrium payoﬀ vector.
Second, Proposition 1 and Theorem 2 suggest that, for any two n-dimensional
monitoring vectors λ and λ, the diﬀerence between the sequential equilibrium payoﬀ
vector sets of Gδ (λ, T ) and Gδ (λ, T ) may disappear if T and δ are large. Proposition 1emphasizes the diﬀerence in the sequential equilibrium payoﬀ vector sets of Gδ (λ, 2)and Gδ (λ, 2). However, Theorem 2 implies that existing folk theorems hold underboth λ and λ. It is an interesting topic for future research to examine whether a very
long horizon and (nearly) complete patience nullify the eﬀects caused by diﬀerences
in monitoring vectors.12

Appendix A.

Proof of Theorem 1

Since Proposition 1 reveals that (B) implies (A), it suﬃces to verify that (A) implies(B).
Fix n ≥ 2 and two n-dimensional monitoring vector λ and λ such that λ ≥ λ.
Also fix an n-player stage game G, δ ∈ (0, 1], and T ≥ 2. Let σ ≡ (σ i )ni=1 be asequential equilibrium of Gδ (λ, T ), let ψ be a consistent system of beliefs supportingσ, and let (σ k )∞k=1 be a tremble of σ such that the sequence of the corresponding
systems of beliefs converges to ψ. For each k, we define a strategy profile of Gδ (λ, T ),σ k ≡ (σ ki )ni=1 , in the following way. For any i such that λi = λi , we define σ ki = σ ki .
For any i such that λi < λi , suppose player i plays Gδ (λ, T ) as follows.
• Player i has a private randomization device, called roulette, which selects 0 withprobability xi ≡ (λi − λi )/(1 − λi ) and selects 1 with probability 1 − xi . Notethat xi ∈ (0, 1).
• In period 1, player i plays the mixed action σ ki prescribes at h1i . If player i
did not obtain automatic information, he spins his roulette. If it selects 0, hemonitors the other players with probability 1. If it selects 1, his monitoring
11

In the stage game constructed in the proof, the unique Nash equilibrium payoﬀ vector is not on
the Pareto frontier of the feasible and individually rational payoﬀ vector set. Therefore, the failureof the existing folk theorems is significant in this game.
12
This issue is absent in [9]. The diﬀerence between purely automatic monitoring and purely
optional monitoring simply enlarges if T gets larger.

13

decision is the one σ ki prescribes at (h1i , ai ), where ai is his stage-game actionin this period.
• Player i’s behavior in period t ≥ 2 depends on his history and past realizationsof his roulette (if any). Suppose t ≥ 2 and he is at a history at period t,τdenoted by hti = (ωiτ )t−1
τ =1 , where ωi ∈ Ii for any τ ≤ t − 1. Define a new historythi = (ω τi )t−1
τ =1 so that for any τ ≤ t − 1,(i) if ωiτ = (a, 1) and if his roulette selected 0 in period τ , ω τi = (a, 0), and(ii) otherwise, ω τi = ωiτ .
t

Let us call hi the eﬀective history. Then, player i plays the mixed action σ kitprescribes at hi . If player i did not obtain automatic information, he spins hisroulette. If it selects 0, he monitors the other players with probability 1. If ittselects 1, his monitoring decision is the one σ ki prescribes at (hi , ai ), where ai is
his stage-game action in this period.
Let us define σ ki as the strategy of Gδ (λ, T ) which is equivalent to the above play.
Note that σ ki is completely mixed because so is σ ki .
Under σ k , whenever a player monitored the other players in a period where hisroulette selected 0, he pretends that automatic information arrived in that period andaccordingly follows σ k . In any period, player i with λi < λi either receives automatic
information or pretends that he received it with probability λi + (1 − λi )xi = λi .
Therefore, σ k generates exactly the same action path as σ k . More precisely, if theplayers play σ k , then
t

t

t

t

(a) for any profile of histories at some period t, h ≡ (hi )ni=1 , the probability thatt
their eﬀective histories at period t are h equals the probability that the playtreaches to h when σ k is played in Gδ (λ, T ), and(b) for any profile of histories at some period t, h ≡ (hi )ni=1 , and any a ∈ A, thetprobability that their eﬀective histories at period t are h and their actions intperiod t are a equals the probability that the play reaches to h and a is playedin period t when σ k is played in Gδ (λ, T ).
Note that these equivalences imply that the payoﬀs of σ k and σ k coincide for any k.
For each k, let ψ k be the system of beliefs derived from σ k by Bayes’ rule. Let(σ, ψ) be the limit of a convergent subsequence of (σ k , ψ k )∞k=1 . By definition, ψ isconsistent given σ. It suﬃces to prove that σ is a sequential equilibrium of Gδ (λ, T ),because σ k → σ, σ k → σ, and continuity imply that the payoﬀs of σ and σ coincide.
Fix a history of player i or his history-action pair arbitrarily, which has the formof either ιti = hti or ιti = (hti , ai ). His continuation strategy under σ i amounts totreplacing hti with the eﬀective history hi and then playing the continuation strategyttof σ i at the new history, which has the form of either ιti = hi or ιti = (hi , ai ). Since σ
is a sequential equilibrium of Gδ (λ, T ), the continuation strategy of σ i at ιti is optimalunder the belief given by ψ. Since σ in Gδ (λ, T ) eﬀectively reproduces σ in Gδ (λ, T ),this implies that the continuation strategy of σ i at ιti when the eﬀective history is ιti
14is optimal under the belief given by ψ. Note that the beliefs under ψ do not dependon the eﬀective history, because any consistent belief does not depend on whether aplayer obtains information by automatic monitoring or own monitoring. This implies
that the continuation strategy of σ i at ιti is optimal under the belief given by ψ,
independently of the eﬀective history. This establishes sequential rationality, and theproof is complete. Q.E.D.

Appendix B.

Proof of Theorem 2

Fix n ≥ 2, and we define G as a special case of the stage game used in the proof of
Proposition 1. Let µ ∈ (0, 1) be a parameter. The sets of each player’s actions are
A1 = {U, M, D}, A2 = {L, C, R}, and if n ≥ 3, Ai = {bi , ci } for any i ≥ 3. As before,
u1 and u2 depend only on (a1 , a2 ), and their payoﬀ matrix is as follows.
L

C

R

U

4, 4

0, 3

µ
4, 3 − 1−µM0, 3 − 1−µ
µ

6, 3

0, 4

D

3, 6

3, 0

3, 6

If n ≥ 3, the payoﬀ function of any player i ≥ 3 is given by (1). As before, G
has a unique Nash equilibrium (U, L, b3 , . . . , bn ). It is easy to show that the full
dimensionality condition holds.
Now, we prove that, for any monitoring vector λ, T ≥ 2 exists such that therepeated game G1 (λ, T ) has a sequential equilibrium payoﬀ vector other than the
Nash equilibrium payoﬀ vector of G.
Fix λ, and choose T ≥ 2 so thatµ ≥ λT1 −1 .

(2)Let σ be the following strategy profile of G1 (λ, T ). For i ≥ 3, player i plays bi at
any history at any period and observes the other players at any history-action pairat any period. Player 1 plays σ1 as follows:• At any history ht1 such that t ≤ T − 2, player 1 plays U , and if the automatic
information did not arrive, he monitors the other players with probability ν,where1µ T −1 − λ1
ν=,1 − λ1(3)

irrespective of his action in period t (note that we have 0 ≤ ν < 1 by (2) andµ < 1).
• In period T − 1, player 1 plays U if he found that player 2 played an action
other than L in some period before period T − 1, and he plays M otherwise.
Whichever action player 1 plays in the period, he observes the other playerswith probability ν, if the automatic information did not arrive.
15

• In period T , player 1 plays M if he found that player 2 played L in all periods
1, 2, . . . , T − 2 and played an action other than C in period T − 1. Otherwise,
player 1 plays U . Then, player 1 observes the other players, irrespective of his
action in that period.
Player 2 plays σ2 as follows:• At any history ht2 such that t ≤ T − 2, player 2 plays L and observes the otherplayers regardless of his action in period t.
• In period T − 1, player 2 plays C if he played L in all past periods. Otherwise,let τ ≥ 1 be the number of past periods when he played an action other thanL. Then(i) if (1 − λ1 )τ (1 − ν)τ ≤ µ, he plays L and observes the other players
irrespective of his action, and(ii) if (1 − λ1 )τ (1 − ν)τ > µ, he plays R and observes the other players
irrespective of his action.
• In period T , player 2 plays C if he played L every period from period 1 toT − 2 and played an action other than C in period T − 1. Otherwise, he plays
L. Then, player 2 observes the other players, irrespective of his action in that
period.
On the path of play, the players play (U, L, b3 , . . . , bn ) in all periods except
period T − 1, and play (M, C, b3 , . . . , bn ) in period T − 1. Hence, the payoﬀ vector of
σ diﬀers from the Nash equilibrium payoﬀ vector of G.
To prove that σ is a sequential equilibrium, we employ a specific tremble andthe corresponding system of beliefs. Let us consider a tremble such that player 2’s
deviation at period T − 1 after playing L in all past periods is far less likely thanany deviation at any other history at any period. Under the corresponding system ofbeliefs, at any history of player 1 at period T such that• in some period t ≤ T − 2, player 1 did not monitor the other players,• player 2 played L in any period t ≤ T − 2 in which player 1 monitored the otherplayers, and• player 1 found that player 2 played an action other than C in period T − 1,player 1 believes that player 2 did not play L in at least one period player 1 did notmonitor the other players.
We show that σ satisfies sequential rationality under this system of beliefs. For
any player i ̸= 2, his action and monitoring decision do not aﬀect the other players’
continuation strategies. Hence, at any history of player i ≥ 3 at any period, it isalways optimal to play bi , a static best reply, and then monitor the others. This provessequential rationality for any player i ≥ 3. For player 1, note that any monitoring
decision is optimal at any history-action pair of player 1 at any period, because theother players’ strategies are pure. Thus, it suﬃces to show that at any history ofplayer 1 at any period, the action σ1 prescribes is his short-run best response given
the belief. First, we consider the histories at period T .
16

1. Player 1 believes that player 2 plays C at any history such that player 1 foundthat player 2 played L in all periods 1, 2, . . . , T − 2 and played an action otherthan C in period T − 1. Hence, M is his short-run best response.
2. Player 1 believes that player 2 plays L at any history such that player 1 foundthat player 2 played an action other than L in some period t ≤ T − 2. Hence,
U is his short-run best response.
3. Player 1 believes that player 2 plays L at any history such that player 1 foundthat player 2 played C in period T − 1. Hence, U is his short-run best response.
4. Consider any history such that (i) player 1 did not monitor the other players inperiod T − 1 and (ii) in any period t ≤ T − 2 in which he monitored the otherplayers, player 2 played L. At that history, player 1 believes that player 2 did
not deviate. Hence, player 1 believes that player 2 plays L in period T , and U
is his short-run best response.
5. Consider any history such that (i) in some period t ≤ T − 2 he did not monitor
the other players, (ii) in any period t ≤ T − 2 in which he monitored theother players, player 2 played L, and (iii) in period T − 1 he found that player 2
played an action other than C. At that history, as we mentioned before, player 1
believes that player 2 did not play L in at least one of the periods in which hedid not monitor the other players. Therefore, player 1 believes that player 2
plays L in period T , and U is his short-run best response.
Next, we consider the histories at period T − 1. When player 1 did not observe a
deviation of player 2, player 1 believes that player 2 plays C in period T − 1. Hence,
M is his short-run best response. When player 1 observed that player 2 deviated insome period before period T − 1, player 1 believes that player 2 never plays C inperiod T − 1. Hence, U is his short-run best response.
Finally, at any history of player 1 at period t ≤ T − 2, player 1 believes that
player 2 plays L in period t. Hence, U is his short-run best response.
Next, we examine sequential rationality of σ2 . Since player 2 is prescribed tomonitor the other players at any history-action pair at any period, his monitoring
decision is always optimal. Let us check the optimality of his actions. We start withthe histories of player 2 at period T , and there are two cases to consider. First, at
any history such that player 2 played L every period from period 1 to T − 2 andplayed an action other than C in period T − 1, player 2 believes that player 1’s actionin period T is M if player 1 monitored the other players in all past periods, and is
U otherwise. The probability with which player 1 monitored the other players in all
past periods is{λ1 + (1 − λ1 )ν}T −1 = µ,where the equality follows from (3). Thus, player 2 believes that player 1 plays U
with probability 1 − µ and M with probability µ in period T . Therefore, it is optimal
for player 2 to play C in period T . Second, at any other history, player 2 believes that
player 1 plays U in period T . Thus, it is optimal for player 2 to play L in period T .
17Let us examine the histories at period T − 1. First, at any history such that
player 2 played L in all past periods, player 2 believes that player 1 plays M for surein period T − 1. If player 2 plays C in period T − 1, his stage payoﬀ is 3. Further, hereaches to a history at period T where he believes that (U, L, b3 , . . . , bn ) is played andreceives the stage payoﬀ of 4. If player 2 does not play C in period T − 1, his stage
payoﬀ is at most 4. Further, he reaches to a history at period T where he believes
that player 1’s action is (1 − µ)U + µM , and therefore his stage payoﬀ is 3. Thus,
the current gain from this deviation is at most 1, but the loss in the next period is
1. Hence, it is optimal for player 2 to play C in period T − 1.
Next, consider a history such that player 2 did not always play L in all past
periods. Let τ ≥ 1 be the number of periods in which he did not play L. Now
player 2 believes that player 1 plays U in period T , irrespective of his action inperiod T − 1. Hence, it suﬃces to show that the prescribed action at this history isa short-run best response of player 2, given his belief. Player 2 believes that player 1
did not observe any deviation with the probability η2 = {(1 − λ1 )(1 − ν)}τ . He thus
expects that player 1’s action in period T − 1 is U with
1 − η2 and M( probability)
1−µwith probability η2 . His stage payoﬀ in period T − 1 is 3 − µ η2 + 4(1 − η2 ) if he()µ
plays L, 3 if he plays C, and 4η2 + 3 − 1−µ (1 − η2 ) if he plays R. If η2 ≤ µ, it is
optimal for player 2 to play L in period T − 1, as is prescribed. Otherwise, playing
R is optimal, again as is prescribed.
Finally, let us fix a history at period t ≤ T − 2. Note first that deviating to an
action other than L in period t reduces his stage payoﬀ by 1 or more. Note also thatwhether he deviated before or not, both conforming to the continuation strategy ofσ2 (and playing C in period T − 1 in case he has not deviated before) and deviatingin the current period make player 1 play U in period T . Thus the only future eﬀect ofa deviation in the current period is to change the outcome in period T − 1. Whether
player 2 deviates or not, he believes that the outcome in period T − 1 is either (i)playing a static best response against player 1 who does not play D and receiving a
stage payoﬀ between 3 and 4, or (ii) playing (M, C, b3 , . . . , bn ) and receiving the stage
payoﬀ of 3. Therefore, the gain in period T − 1 when player 2 deviated in period t is
at most one. This establishes that no one-shot deviation at this history pays, which
completes the proof. Q.E.D.

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