Repeated games: Overlapping generations Felix Munoz-Garcia Strategy and Game Theory - Washington State University
Chapter 15 Chapter 15 (Harrington) - Cooperation in in…nitely lived 1 institutions. So far individuals interacting in an in…nitely repeated game 2 knew there were some chances they were going to meet each other again. i.e., cooperation was sustained by the "shadow of the future" 1 hanging over future encounters. But in some cases individuals know for sure they won’t see 3 each other again. Why do people cooperate then? 1 In this chapter we will examine cooperation in institutions 4 where individuals are …nitely lived, but 1 the institution lasts forever. 2
Chapter 15 An in…nitely lived institution can be understood as an 1 overlapping generations model in macroeconomics. That is, at any stage some people are young, some are 1 middle-aged, some are old. Importantly, when the old die in the following period, the 2 population is replenished by newborns. Hence, the institution lives forever. 3 How can we sustain cooperation in these settings? 2
Chapter 15 Another potentially problematic setting: 1 People interact only one period: Businessmen A and B meet 1 only once. If I am businessman A, how I am going to discipline B (playing 2 a punishment strategy, as in the GTS) if I never meet businessman B again? Although one person cannot discipline another, society at large 3 might be able to perform that function. For example, if information about past encounters is observed 4 by other people who will interact with businessman B in the future, they can punish him for acting improperly towards A. We will describe how to sustain cooperation in these settings. 2
Overlapping generations and tribal defense Consider a nation, a village, or tribe with N � 2 members. 1 Each member decides: 2 whether to exert e¤ort defending the group (public project), at 1 a private cost of 10, or shirk. 2 Every member obtains a bene…t of 6 units for every individual 3 who exerts e¤ort. Hence, if m members exert e¤ort, my utility is 4 8 Me! Cost < z}|{ z}|{ u i ( s i , m ) = : 6 ( m + 1 ) � 10 if s i = exert e¤ort 6 m if s i = no e¤ort
Overlapping generations and tribal defense Given utility 1 � 6 ( m + 1 ) � 10 if s i = exert e¤ort u i ( s i , m ) = 6 m if s i = no e¤ort it is immediate to show that exering e¤ort is a strictly dominated strategy. In particular, 2 6 ( m + 1 ) � 10 < 6 m ( ) 6 m � 4 < 6 m ( ) � 4 < 0 which holds for any value of m . That is, I have incentives to free-ride (shirk) regardless of the 3 number of individuals who end up exerting e¤ort. Hence, the psNE of the unrepeated game has s i = no e¤ort for 4 every player i 2 N .
Overlapping generations and tribal defense For simplicity, let’s solve this game as we know so far: when 1 players interact in…nitely often (they never die). In this case, we can design the following modi…ed GTS:. 2 At t = 1, exert e¤ort (cooperate) 1 At t > 1, exert e¤ort if all players exerted e¤ort in all previous 2 periods... but temporarily revert to no e¤ort for one period if any player 1 deviates from exerting e¤ort in previous periods. Then, after one period of reversion (punishment), go back to 2 the cooperative outcome, i.e., exert e¤ort.
Overlapping generations and tribal defense After a history of cooperation, my payo¤ if I keep cooperating 1 is: ( 6 N � 10 ) + δ ( 6 N � 10 ) + δ 2 ( 6 N � 10 ) + ... While my payo¤ from deviating to no e¤ort is: 2 + δ 2 ( 6 N � 10 ) 6 ( N � 1 ) + δ 0 + ... |{z} | {z } | {z } punishment in psNE you are not coop go back to coop while all other ( N � 1 ) members cooperate
Overlapping generations and tribal defense Comparing these payo¤s, cooperation can be sustained as the 1 SPNE of the in…nitely repeated game if: ( 6 N � 10 ) + δ ( 6 N � 10 ) + (((((((( ( δ 2 ( 6 N � 10 ) + ... ( 6 ( N � 1 ) + δ 0 + (((((((( δ 2 ( 6 N � 10 ) + ... � Rearranging, 6 N � 10 + δ ( 6 N � 10 ) � 6 N � 6 Hence, 4 δ � 6 N � 10 Figure of this cuto¤ for δ (next slide) 2
Overlapping generations and tribal defense Minimal discount factor supporting cooperation in the Overlapping Generation-Tribal defense game, as a function of the population size, N δ 0. 008 0. 006 Coop 0. 004 0. 002 Do not coop 4 δ = 6 N - 10 N 200 400 600 800 1000 Cooperation is easier to sustain the larger the population is.
Overlapping generations and tribal defense But, what if players do not interact in…nitely often ? 1 You live during T periods only, and there are N members in 2 total. At any period T , there are N T members currently alive in this 3 generation T. Example : N = 100 and T = 4 years, then N T = 100 = 25 1 4 members are children, 25 are teenagers, 25 are adults, and 25 are seniors.
Overlapping generations and tribal defense Then, at any period, 1 N � N T people are younger than age T � T � 1 � = NT � N = N T T In the previous example where N = 100 and T = 4 years, 1 � � � � T � 1 3 � 1 N = 100 = 75 individuals are younger than the T 4 maximum age any member in the population reaches. In particular, 25 members are children, 25 are teenagers, and 2 25 are adults.
Overlapping generations and tribal defense Let us now analyze how to support cooperation in this setting. 1 Consider the following strategy 2 At the last period of your life (period T ), you don’t exert any 1 e¤ort (e.g., retirement for seniors). How would I be disciplined otherwise? When we meet them in 1 the afterlife? During all previous T � 1 periods, you exert e¤ort, but if 2 someone deviates from this strategy: you revert to the psNE of the stage game during one period 1 (temporary punishment), and move to the cooperative outcome (exerting e¤ort) afterwards. 2
Overlapping generations and tribal defense In order to show that such strategy can be sustained as SPNE 1 of the game, we must show that it is optimal for: the individual who is in the last period ( T ) of his life (of 1 course!). the individual who is in the penultimate period ( T � 1) of his 2 life. the individual who is in period T � 2 of his life. 3 the individual who is in period T � 3 of his life, etc. 4
Overlapping generations and tribal defense Payo¤ in penultimate period of life, i.e., T � 1: Payo¤ from cooperating: 2 3 2 3 � T � 1 � � T � 1 � 6 7 6 7 6 7 6 7 4 6 N � 10 + δ 4 6 N 5 5 T T | {z } | {z } m m | {z } | {z } e¤ort no e¤ort, he is a senior but m is una¤ected, thanks to the newborns! Payo¤ from deviating: 2 3 � T � 1 � 6 7 6 7 6 6 N � 1 5 + 7 δ � 0 |{z} 4 T | {z } punished during retirement! m , without you
Overlapping generations and tribal defense Comparing, � T � 1 � � � � T � 1 � � � T � 1 � � ������ ������ 6 N � 10 + δ 6 N � 6 N � 6 T T T 4 2 � T � 1 � � T � 1 � δ � N = (Condition 1) 6 3 N T T
Overlapping generations and tribal defense Player who is still two periods from retirement, i.e., T � 2 (Teenager): Payo¤ from cooperation: � � T � 1 � � � � T � 1 � � � � T � 1 � � + δ 2 6 N � 10 + δ 6 N � 10 6 N T T T | {z } | {z } | {z } e¤ort as a teenager e¤ort as an adult no e¤ort as a senior Payo¤ from deviating to no e¤ort: �� T � 1 � � � � T � 1 � � + δ 2 6 N � 1 + δ � 0 6 N |{z} T T | {z } | {z } punished as an adult... I shirk as a teenager... but enjoy life as a senior!
Overlapping generations and tribal defense Let’s compare the payo¤s. First, note that last period payo¤s were the same. Hence, we don’t even write them in our payo¤ comparison. � � � T � 1 � � � T � 1 � � � T � 1 � ������ ������ 6 N � 10 + δ 6 N � 10 � 6 N � 6 T T T � � T � 1 � � = ) δ 6 N � 10 � 10 � 6 T 4 � T � 1 � = ) δ � 6 N � 10 T 2 = � T � 1 � (Condition 3) 3 N � 5 T
Overlapping generations and tribal defense Similarly for individuals in previous periods, e.g., T � 3: Payo¤ from cooperation: � � T � 1 � � � � T � 1 � � 6 N � 10 + δ 6 N � 10 T T | {z } | {z } e¤ort as a child e¤ort as a teenager � � T � 1 � � � � T � 1 � � + δ 2 + δ 3 N � 10 6 6 N T T | {z } | {z } e¤ort as an adult no e¤ort as a senior
Overlapping generations and tribal defense Payo¤ from deviating to no e¤ort: �� T � 1 � � 6 N � 1 + δ � 0 |{z} T | {z } punished as a teenager I shirk as a child � � T � 1 � � � � T � 1 � � + δ 2 + δ 3 6 N � 10 6 N T T | {z } | {z } e¤ort as an adult no e¤ort as a senior
Overlapping generations and tribal defense Comparing Note that the last two period payo¤s were the same. Hence, we don’t need to write it down in our payo¤ comparison. � T � 1 � � � � T � 1 � � � T � 1 � � ������ ������ 6 N � 10 + δ 6 N � 10 � 6 N � 6 T T T � � T � 1 � � = ) δ 6 N � 10 � 10 � 6 T 4 2 � T � 1 � � T � 1 � = ) δ � N � 10 = 6 3 N � 5 T T (Coincides with our above Coindition 2)
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