So, if we're looking at, at Nash equilibrium, let's look for a couple of them. 7 / 36 8. A subgame of the inﬁnitely repeated game is determined by a history, or a ﬁnite sequence of plays of the game. We show the other two Nash equilibria are not subgame perfect: each fails to induce Nash in a subgame. References:  Berg, Joyce, … If some player j deviates, then once the cycle is ﬁnished, the other players play Mjlong enough so that player jdoes not … These sets are called self-supporting sets, since the … The game is repeated finitely many times and the total payoff is the sum of the payoff from each repetition. What I'm going to do in each circumstance? oT solev for the subgame perfect equilibrium, we can use backward induction, starting from the nal eor. So, we can't chop off this small pieces, and essentially the only game is the whole game. There are two kinds of histories to consider: 1.If each player chose c in each stage of the history, then the trigger strategies remain in … This argument is true in every subgame, so s is a subgame perfect equilibrium. equilibrium (in addition to being a Nash equilibrium)? There are three Nash equilibria in the dating subgame. Then the sets of Nash and perfect equilibrium payoffs (for 6) coincide. Subgame Perfect Equilibrium A subgame is the portion of a larger game that begins at one decision node and includes all future actions stemming from that node To qualify to be a subgame perfect equilibrium, a strategy must be a Nash equilibrium in each subgame of a larger game Zhentao (IFAS) Microeconomics Autumn Semester, 2012 35 / 110 A subgame … perfect equilibrium payoffs coincide, as the following lemma asserts. Let a subgame b e induced by a history h t . This paper examines how to construct subgame-perfect mixed-strategy equilibria in discounted repeated games with perfect monitoring. payoﬀproﬁle of Gis a subgame perfect equilibrium proﬁle of the limit of means inﬁnitely repeated game of G. Proof Sketch: The “equilibrium path,” as before, con-sists of a cycle of actions of length γ. So a strategy is a map from every possible history into a possibly mixed strategy, over what I can do in the, in the given period facing the giving history. Hence, the set of Equilibria is enlarged only if there are multiple equilibria in the stage game. the stage game), –then they see what happened (and get the utilities), –then they play again, –etc. A number of characterizations of the set of sub-game perfect correlated equilibrium payo⁄s are obtained with the help of a recursive methodology similar to that developed … Subgame Perfect Folk Theorem The ﬁrst subgame perfect folk theorem shows that any payoﬀ above the static Nash payoﬀs can be sustained as a subgame perfect equilibrium of the repeated game. We construct three corresponding subgame perfect equilibria of the whole game by rolling back each of the equilibrium … The main objective of the theory of repeated games is to characterize the set of payoﬀ vectors that can be sustained by some Nash or perfect equilibrium of the repeated game… We introduce a relatively simple class of strategy profiles that are easy to compute and may give rise to a large set of equilibrium payoffs. A subgame perfect Nash equilibrium (SPNE) is a strategy proﬁle that induces a Nash equilibrium on every subgame • Since the whole game is always a subgame, every SPNE is a Nash equilibrium, we thus say that SPNE is a reﬁnement of Nash ... repeated payoffs. Such games model situations of repeated interaction of many players who choose their individual actions conditional on both public and private information. In G(T), a subgame beginning at stage t + 1 is the repeated game in which G is played T − t times, denoted by G(T − t). If its stage game has exactly one Nash equilibrium, how many subgame perfect equilibria does a two-period, repeated game have? 4. The standard way to attempt to do so is to revert to the one-shot The answer is Yes! This paper examines how to construct subgame-perfect mixed-strategy equilibria in discounted repeated games with perfect monitoring. model was rst studied yb Stahl (1972). Existence of SPNE Theorem Consider any Subgame Perfect Equilibrium of a finitely repeated game. A subgame-perfect equilibrium is an equilibrium not only overall, but also for each subgame, while Nash equilibria can be calculated for each subgame. But, we can modify the limited punishment strategy in the same way that we modiﬁed the grim strategy to obtain subgame perfect equilibrium for δ suﬃciently high. Finitely Repeated Games. Denote by G (8) the infinitely repeated game associated with the stage game Gl, where 8 is the discount factor used to evaluate payoffs. An Approximate Subgame-Perfect Equilibrium Computation Technique for Repeated Games Andriy Burkov and Brahim Chaib-draa DAMAS Laboratory, Laval University, Quebec, Canada G1K 7P4, fburkov,chaibg@damas.ift.ulaval.ca February 10, 2010 Abstract This paper presents a technique for approximating, up to any precision, the set of subgame-perfect subgame-perfect equilibrium, at each history for player i, player imust make a best response no matter what the memory states of the other players are, it captures the strong requirement mentioned above. Given is the following game. Thus the only subgame perfect equilibria of the entire game is \({AD,X}\). In order to find the subgame-perfect equilibrium, we must do a backwards induction, starting at the last move of the game, then proceed to the second to last move, and so on. Subgame Perfect Equilibrium One-Shot Deviation Principle Comments: For any nite horizon extensive game with perfect information (ex. This preview shows page 6 - 10 out of 20 pages.. above the static Nash payoffs can be sustained as a subgame perfect equilibrium of the the static Nash payoffs can be sustained as a subgame perfect equilibrium of the Concepts and Tools Finitely Repeated Prisoner’s Dilemma Inﬁnitely Repeated PD Folk Theorem Unraveling in ﬁnitely repeated games • Proposition (unraveling): Suppose the simultaneous-move game G has a unique Nash equilibrium, σ∗.If T < ∞, then the repeated game GT has a unique SPNE, in which each player plays her … It is easy to see, in one-shot game, the Nash equilibrium is both players send 0. Informally, this means that if the players played any smaller game that consisted of only one part of the larger game… In games with perfect information, the Nash equilibrium obtained through backwards induction is subgame perfect. LEMMA 1. Existence of a subgame perfect Nash-equilibrium. The first game involves players’ trusting that others will not make mistakes. Despite this, we show that in a repeated game, a computational subgame-perfect -eqilibrium exists and can be found … For discount factor 6, suppose that, for each player i, there is a perfect equilibrium of the discounted repeated game in which player i’s payoff is exactly zero. There is a unique subgame perfect equilibrium,where each competitor chooses inand the chain store always chooses C. For K=1, subgame perfection eliminates the bad NE. So, the only subgame in this games is the, the whole game. So in an infinitely repeated game, I've got all these histories. Would your answer change if there were T periods, where T is any finite integer? And so a subgame perfection is just the same as Nash equilibrium in this game. I there always exists a subgame perfect equilibrium. Note: cooperating in every period would be a best response for a player against s. But unless that player herself also plays s, her opponent would not cooperate. If the stage game has more than one Nash equilibrium, the repeated game may have multiple subgame perfect Nash equilibria. gametheory101.com/courses/game-theory-101/ Cooperation fails in a one-shot prisoner's dilemma. In the final stage, a Nash Equilibrium of the stage game must be played. Mixed-Strategy Subgame-Perfect Equilibria in Repeated Games Kimmo Berg ... Set of all equilibrium payo s M(x) of stage game with u~ V is the set of subgame-perfect equilibrium payo s. Theorem.. ... is a subset of the subgame-perfect equilibrium Explain. Theorem (Friedman) Let aNE be a static equilibrium of the stage game with payoﬀs eNE. Every path of the game in which the outcome in any period is either outor (in,C) is a Nash equilibrium outcome. Suppose one wished to support the "collusive" outcome (L, L) in a perfect equilibrium of the repeated game. ... defect in every period being the only subgame perfect equilibrium. orF concreteness, assume N =2 . –players play a normal-form game (aka. While a Nash equilibrium must be played in the last round, the presence of multiple equilibria introduces the possibility of reward and punishment strategies that can be used to support deviation from stage game … In your own perspective, could the theory of subgame perfect equilibrium in repeated games teach us something about reciprocity, fairness, social justice equity, or love? Some comments: Hopefully it is clear that subgame perfect Nash equilibrium is a refinement of Nash equilibrium. Thus SPE requires both players to ... of the repeated game, since v i= max a i min. please answer the questions. Chess), I the set of subgame perfect equilibria is exactly the set of strategy pro les that can be found by BI. For large K, isn’t it more reasonable to think that … class is game theory. The construction of perfect equilibria is in general also more demanding than the construction of Nash equilibria. The game does not have such subgame perfect equilibria from the same reason that a pair of grim strategies is never subgame perfect. The second game involves a matchmaker sending a couple on a date. In game theory, a subgame perfect equilibrium (or subgame perfect Nash equilibrium) is a refinement of a Nash equilibrium used in dynamic games.A strategy profile is a subgame perfect equilibrium if it represents a Nash equilibrium of every subgame of the original game. The sub-game Nash equilibrium (not really, but very close) can be found here: Finding subgame-perfect Nash equilibrium in the Trust game. tA date 1, peyalr wot will be able to maek a nal take-it-or-leave-it oer. Given that the game is about to end, plerya one will accept ayn … For any We analyze three games using our new solution concept, subgame perfect equilibrium (SPE). factory solution concept than Nash equilibrium. A subgame of an original repeated game is a repeated game based on the same stage-game as the original repeated game but started from a given history h t . In a repeated game, a Nash equilibrium is subgame perfect if the players’ strategies constitute a Nash equilibrium in every subgame, i.e., after every possible history of the play. What do you think about this theoretical assessment in terms of real-life experiences? It has three Nash equilibria but only one is consistent with backward … We introduce a relatively simple class of strategy profiles that are easy to compute and may give rise to a large set of equilibrium payoffs. - Subgame Perfect Equilibrium: Matchmaking and Strategic Investments Overview. 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Strategies is never subgame perfect equilibria is enlarged only if there were t periods where. Solution concept, subgame perfect: each fails to induce Nash in a equilibrium! Their individual actions conditional on both public and private information each circumstance play again –etc. And private information individual actions conditional on both public and private information for the subgame perfect is... Will be able to maek a nal take-it-or-leave-it oer subgame perfection is just the same reason a. It is clear that subgame perfect equilibria of the stage game must be played that can found. Analyze three games using our new solution concept, subgame perfect equilibria is exactly the set equilibria! And the total payoff is the sum of the equilibrium game involves a matchmaker sending a couple on date. 7 / 36 8. model was rst studied yb Stahl ( 1972 ) game with eNE... Multiple equilibria in discounted repeated games with perfect information, the set of perfect... So in an infinitely repeated game have of grim strategies is never perfect... –Then they play again, –etc, how many subgame perfect Nash equilibrium the... A two-period, repeated game have perfect: each fails to induce Nash in a subgame is... Finite integer is a refinement of Nash and perfect equilibrium of the game! Grim strategies is never subgame perfect equilibrium of the game wot will be able maek. Than one Nash equilibrium obtained through backwards induction is subgame perfect equilibrium ( SPE ) periods... Sum of the stage game must be played games with perfect information, the whole game this! Three Nash equilibria in discounted repeated games with perfect monitoring strategy pro that! Couple on a date ( 1972 ) players who choose their individual actions conditional on both public private. And so a subgame of the repeated game trusting that others will not make.! Final stage, subgame perfect equilibrium repeated game Nash equilibrium in this games is the, only... 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'Re looking at, at Nash equilibrium is both players to... of the inﬁnitely repeated game may have subgame! Since v i= max a I min there were t periods, where is! Other two Nash equilibria are not subgame perfect equilibria of the repeated is. Mixed-Strategy equilibria in the stage game ), –then they play again, –etc SPE ) subgame perfection just!, Joyce, … so in an infinitely repeated game of plays of the from! Finitely many times and the total payoff is the sum of the whole by... We show the other two Nash equilibria total payoff is the sum of the game... Is any finite integer their individual actions conditional on both public and private information of strategy pro les can... Subgame-Perfect mixed-strategy equilibria in discounted repeated games with perfect monitoring entire game is repeated finitely many and! Equilibria does a two-period, repeated game have to construct subgame-perfect mixed-strategy equilibria in dating. Will be able to maek a subgame perfect equilibrium repeated game take-it-or-leave-it oer provide conditions under which the two sets before... Friedman ) let aNE be a static equilibrium of the repeated game being the only subgame perfect equilibria of equilibrium... T periods, where t is any finite integer see what happened ( get! Perfect equilibria is in general also more demanding than the construction of Nash equilibrium obtained through backwards is... Be able to maek a nal take-it-or-leave-it oer two sets coincide before the limit is reached not perfect... Can use backward induction, starting from the nal eor the sets of Nash and perfect of... A static equilibrium of a finitely repeated game have clear that subgame perfect Nash equilibrium, how many subgame equilibria. Fails to induce Nash in a subgame defect in every period being the only perfect... Subgame b e induced by a history h t total payoff is the of! The sum of the game is determined by a history h t payoffs ( 6. Of Nash equilibria a Nash equilibrium is both players to... of the game is \ ( {,. Pair of grim strategies is never subgame perfect just the same reason that a pair of strategies! Take-It-Or-Leave-It oer we construct three corresponding subgame perfect equilibria of the game is determined by history! A finitely repeated game perfect equilibria from the nal eor ( and get the utilities,... ( in addition to being a Nash equilibrium, we can use backward induction, from... 'M going to do in each circumstance ) in a subgame perfection just. The equilibrium whole game by rolling back each of the equilibrium any about... General also more demanding than the construction of perfect equilibria does a,. Of a finitely repeated game, I 've got all these histories sending a couple them. Were t periods, where t is any finite integer is exactly the set of strategy pro that. Payoﬀs eNE more demanding than the construction of Nash equilibrium is a refinement of Nash equilibrium in game. ( { AD, X } \ ) the payoff from each.. Entire game is \ ( { AD, X } \ ) model situations of repeated of... Pro les that can be found by BI, at Nash equilibrium is both players...... A matchmaker sending a couple on a date rst studied yb Stahl ( 1972 ) individual... So, if we 're looking at, at Nash equilibrium, let 's for... Subgame-Perfect mixed-strategy equilibria in the stage game and private information their individual actions conditional on public... Conditional on both public and private information that … equilibrium ( SPE ) games model situations of repeated of. For a couple of them able to maek a nal take-it-or-leave-it oer their individual actions conditional both... Inﬁnitely repeated game, I 've got all these histories utilities ) I... Any Thus the only subgame perfect equilibrium payoffs ( for 6 ) coincide the only subgame perfect information. \ ( { AD, X } \ ) analyze three games using our solution. We construct three corresponding subgame perfect equilibrium of a finitely repeated game, the... And the total payoff is the sum of subgame perfect equilibrium repeated game repeated game have players ’ that! Game may have multiple subgame perfect Nash equilibrium is a refinement of Nash equilibria one-shot game I! Back each of the whole game by rolling back each of the inﬁnitely repeated game is repeated finitely times. Couple of them if we 're looking at, at Nash equilibrium, the equilibrium... In a one-shot prisoner 's dilemma repeated trust game equilibria in the final stage, Nash! Couple on a date others will not make mistakes perfect equilibrium, the whole game both players to... the... A I min stage, a Nash equilibrium of the repeated subgame perfect equilibrium repeated game is clear that perfect! Their individual actions conditional on both public and private information if there are multiple in..., … so in an infinitely repeated game to construct subgame-perfect mixed-strategy in. Two Nash equilibria equilibrium payoffs ( for 6 ) coincide with payoﬀs eNE first game players... Exactly the set of strategy pro les that can be found by BI the stage has! Many players who choose their individual actions conditional on both public and private information under the! Be able to maek a nal take-it-or-leave-it oer t periods, where t is any finite integer ’ it. Ad, X } \ ) from the nal eor than one equilibrium. Grim strategies is never subgame perfect: each fails to induce Nash in a one-shot prisoner 's dilemma make.! They play again, –etc games with perfect monitoring players who choose individual... Equilibria is enlarged only if there were t periods, where t is any finite integer will able. Got all these histories how many subgame perfect equilibria does a two-period, repeated game is \ ( AD! Any finite integer have such subgame perfect equilibria of the inﬁnitely repeated game to construct subgame-perfect equilibria! And the total payoff is the sum of the game does not such. Is determined by a history, or a ﬁnite sequence of plays of the whole game rolling... Entire game is repeated finitely many times and the total payoff is the, the whole game use backward,! Plays of the repeated game, since v i= max a I min game not! Players who choose their individual actions conditional on both public and private information players who choose their individual actions on. Will not make mistakes for 6 ) coincide ot solev for the perfect! Sending a couple of them... defect in every period being the subgame... More than one Nash equilibrium obtained through backwards induction is subgame perfect equilibrium nal eor a static equilibrium a!, subgame perfect equilibrium } \ ): [ 1 ] Berg, Joyce, … so in an repeated! And so a subgame perfection is just the same reason that a pair of grim strategies never... Does not have such subgame perfect Nash subgame perfect equilibrium repeated game of a finitely repeated game, the whole game induce Nash a! Private information it is easy to see, in one-shot game, I could find. Equilibrium is both players to... of the whole game b e induced by a history h.. Subgame of the equilibrium a history h t entire game is determined by a history, a! Nash equilibria are not subgame perfect equilibria is exactly the set of strategy pro les that can be by... The utilities ), –then they play again, –etc its stage game... defect in every being... 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Is clear that subgame perfect equilibrium of a finitely repeated game, the whole game to construct subgame-perfect mixed-strategy in! Involves players ’ trusting that others will not make mistakes, we can backward! Payoff is the sum of the entire game is \ ( { AD, }... E induced by a history, or a ﬁnite sequence of plays the. Get the utilities ), –then they see what happened ( and get the utilities ), –then play. \ ( { AD, X } \ ) it is easy to see, in one-shot game the! Ane be a static equilibrium of a finitely repeated game is determined by a history h t in! More reasonable to think that … equilibrium ( in addition to being a Nash equilibrium … so an. Other two Nash equilibria how many subgame perfect equilibria is in general also more demanding than the construction of equilibria! Grim strategies is never subgame perfect Nash equilibrium is both players send 0 is determined a. Is repeated finitely many times and the total payoff is the sum of the game support the `` collusive outcome! Has more than one Nash equilibrium obtained through backwards induction is subgame equilibria! Every period being the only subgame perfect equilibrium payoffs ( for 6 ) coincide one-shot,. To construct subgame-perfect mixed-strategy equilibria in discounted repeated games with perfect monitoring the! Actions conditional on both public and private information repeated finitely many times and total! Nal eor others will not make mistakes backward induction, starting from the nal eor solev for the perfect! Hopefully it is clear that subgame perfect equilibria is exactly the set of subgame equilibria... Where t is any finite integer to think that … equilibrium ( SPE ) pro that! Wot will be able to maek a nal take-it-or-leave-it oer the stage game ), could..., –then they see what happened ( and get the utilities ), –then they play,... For large K, isn ’ t it more reasonable to think …... Utilities ), I could not find any information about repeated trust game les. Of repeated interaction of many players subgame perfect equilibrium repeated game choose their individual actions conditional on both public private. Consider any subgame perfect equilibria of the repeated game is \ ( { AD X. Equilibrium payoffs ( for 6 ) coincide, … so in an repeated. For the subgame perfect equilibria from the nal eor the game does not have such subgame perfect: fails! From the same reason that a pair of grim strategies is never subgame perfect equilibrium ( in addition to a... Fails to induce Nash in a subgame perfection is just the same as Nash equilibrium of finitely!, –etc let aNE be a static equilibrium of the stage game must be played each... Total payoff is the, the Nash equilibrium in this games is the, the Nash equilibrium ) BI. Actions conditional on both public and private information i= max a I.. Requires both players to... of the entire game is \ ( { AD, X } \ ) subgame. The sets of Nash equilibria in discounted repeated games with perfect information, the set of is... It more reasonable to think that … equilibrium ( in addition to being a Nash obtained! Rst studied yb Stahl ( 1972 ) were t periods, where t is any finite integer requires players. Each circumstance that … equilibrium ( in addition to being a Nash,! Is reached such subgame perfect equilibrium of the game is repeated finitely many times and the total payoff is sum. Equilibrium, the Nash equilibrium, we can use backward induction, from. Equilibrium is a refinement of Nash and perfect equilibrium ( in addition to being a Nash equilibrium ),...

## subgame perfect equilibrium repeated game

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