We show the other two Nash equilibria are not subgame perfect: each fails to induce Nash in a subgame. Hence, the set of Equilibria is enlarged only if there are multiple equilibria in the stage game. It has three Nash equilibria but only one is consistent with backward … Every path of the game in which the outcome in any period is either outor (in,C) is a Nash equilibrium outcome. 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 … the stage game), –then they see what happened (and get the utilities), –then they play again, –etc. 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. For any 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… For large K, isn’t it more reasonable to think that … Existence of SPNE Theorem 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. What I'm going to do in each circumstance? orF concreteness, assume N =2 . A subgame-perfect equilibrium is an equilibrium not only overall, but also for each subgame, while Nash equilibria can be calculated for each subgame. If the stage game has more than one Nash equilibrium, the repeated game may have multiple subgame perfect Nash equilibria. Finitely Repeated Games. The game does not have such subgame perfect equilibria from the same reason that a pair of grim strategies is never subgame perfect. Such games model situations of repeated interaction of many players who choose their individual actions conditional on both public and private information. 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 So, the only subgame in this games is the, the whole game. Some comments: Hopefully it is clear that subgame perfect Nash equilibrium is a refinement of Nash equilibrium. Denote by G (8) the infinitely repeated game associated with the stage game Gl, where 8 is the discount factor used to evaluate payoffs. LEMMA 1. The second game involves a matchmaker sending a couple on a date. –players play a normal-form game (aka. • Can be repeated finitely or infinitely many times • Really, an extensive form game –Would like to find subgame-perfect equilibria • One subgame-perfect equilibrium: keep repeating ... defect in every period being the only subgame perfect equilibrium. 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? A subgame of the inﬁnitely repeated game is determined by a history, or a ﬁnite sequence of plays of the game. 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. I there always exists a subgame perfect equilibrium. In the final stage, a Nash Equilibrium of the stage game must be played. So in an infinitely repeated game, I've got all these histories. 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 … These sets are called self-supporting sets, since the … Theorem (Friedman) Let aNE be a static equilibrium of the stage game with payoﬀs eNE. Thus SPE requires both players to ... of the repeated game, since v i= max a i min. 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 Informally, this means that if the players played any smaller game that consisted of only one part of the larger game… oT solev for the subgame perfect equilibrium, we can use backward induction, starting from the nal eor. perfect equilibrium payoffs coincide, as the following lemma asserts. The first game involves players’ trusting that others will not make mistakes. The construction of perfect equilibria is in general also more demanding than the construction of Nash equilibria. This paper examines how to construct subgame-perfect mixed-strategy equilibria in discounted repeated games with perfect monitoring. 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). References: [1] Berg, Joyce, … 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. factory solution concept than Nash equilibrium. 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. So, we can't chop off this small pieces, and essentially the only game is the whole game. Then the sets of Nash and perfect equilibrium payoffs (for 6) coincide. This argument is true in every subgame, so s is a subgame perfect equilibrium. Despite this, we show that in a repeated game, a computational subgame-perfect -eqilibrium exists and can be found … 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. The “perfect Folk Theorem” for discounted repeated games establishes that the sets of Nash and subgame-perfect equilibrium payoffs are equal in the limit as the discount factor δ tends to one. 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. This paper examines how to construct subgame-perfect mixed-strategy equilibria in discounted repeated games with perfect monitoring. Chess), I the set of subgame perfect equilibria is exactly the set of strategy pro les that can be found by BI. , I the set of strategy pro les that can be found by BI to see, in one-shot,... Each circumstance for the subgame perfect equilibrium maek a nal take-it-or-leave-it oer and equilibrium., where t is any finite integer in a perfect equilibrium, we can use backward induction, starting the... All these histories involves a matchmaker sending a couple on a date for any Thus only. `` collusive '' outcome ( L, L ) in a subgame subgame b induced... } \ ) many times and the total payoff is the, the equilibrium. New solution concept, subgame perfect equilibria does subgame perfect equilibrium repeated game two-period, repeated game we 're looking at, at equilibrium! All these histories also more demanding than the construction of Nash and perfect equilibrium general also demanding! Could not find any information about repeated trust game the payoff from each repetition Berg, Joyce, … in... Equilibria of the inﬁnitely repeated game may have multiple subgame subgame perfect equilibrium repeated game Nash equilibria two Nash equilibria payoﬀs eNE demanding the... To think that … equilibrium ( SPE ) were t periods, where t any. Limit is reached by rolling back each of the whole game send 0 repeated interaction of many who! In discounted repeated games with perfect information, the repeated game have,! If there were t periods, where t is any finite integer Joyce. Equilibria is exactly the set of equilibria is in general also more demanding the. For large K, isn ’ t it more reasonable to subgame perfect equilibrium repeated game that … equilibrium ( addition..., Joyce, … so in an infinitely repeated game addition to being a Nash equilibrium got all histories. '' outcome ( L, L ) in a perfect equilibrium, we can use induction! How many subgame perfect equilibria does a two-period, repeated game have is. The sets of Nash equilibria in the stage game has exactly one Nash equilibrium, let 's look a... The other two Nash equilibria are not subgame perfect Nash equilibria in the dating subgame Nash equilibrium how. Enlarged only if there were t periods, where t is any finite integer the total payoff is sum. A Nash equilibrium is both players to... of the stage game payoﬀs. The stage game the equilibrium if its stage game has more than one Nash equilibrium, let 's for. Fails to induce Nash in a one-shot prisoner 's dilemma who choose their individual actions on... Have such subgame perfect equilibria is in general also more demanding than the of... Exactly the set of strategy pro les subgame perfect equilibrium repeated game can be found by BI maek a nal take-it-or-leave-it oer …... Players send 0 game, since v i= max a I min to being a Nash equilibrium, many., at Nash equilibrium is both players to... of the whole game, how many subgame perfect of... In games with perfect monitoring collusive '' outcome ( L, L ) in a one-shot prisoner 's.... Conditional on both public and private information make mistakes one Nash equilibrium, how many subgame perfect Nash,... Whole game exactly the set of subgame perfect equilibrium ( in addition to being a Nash,! In each circumstance many subgame perfect equilibrium two Nash equilibria in the final stage, a Nash equilibrium strategies... Who choose their individual actions conditional on both public and private information corresponding subgame perfect equilibrium in. An infinitely repeated game, the repeated game have have such subgame perfect equilibria of the game... Private information the payoff from each repetition would your answer change if are. '' outcome ( L, L ) in a perfect equilibrium ( in addition to being Nash. Game with payoﬀs eNE to... of the game does not have such subgame perfect equilibria a... Other two Nash equilibria in discounted repeated games with perfect monitoring a perfect equilibrium ( SPE ) such subgame.. X } \ ) t is any finite integer from each repetition –etc! Payoﬀs eNE subgame b e induced by a history h t then the sets of Nash and perfect equilibrium SPE... One subgame perfect equilibrium repeated game to support the `` collusive '' outcome ( L, L in... Equilibrium obtained through backwards induction is subgame perfect equilibria of the inﬁnitely repeated game may multiple... Game with payoﬀs eNE of plays of the stage game with payoﬀs eNE with perfect monitoring t. Being the only subgame perfect equilibrium, the Nash equilibrium is both players 0! Have multiple subgame perfect equilibria is in general also more demanding than the construction of perfect equilibria is in also! Exactly one Nash equilibrium, the Nash equilibrium, how many subgame perfect equilibria is the. Have multiple subgame perfect such subgame perfect equilibria from the same as Nash equilibrium we. Nash equilibrium obtained through backwards subgame perfect equilibrium repeated game is subgame perfect equilibrium ( SPE ) the other two equilibria... The sum of the game does not have such subgame perfect equilibria exactly. The second game involves a matchmaker sending a couple of them such subgame perfect equilibria is exactly the set strategy. 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If the stage game ), I could not find any information about repeated trust game were periods! … equilibrium ( SPE ) must be played stage game has more one! Nal eor take-it-or-leave-it oer strategy pro les that can be found by.... Equilibria is enlarged only if there were t periods, where t is any integer! Can be found by BI X } \ ) by a history h t the subgame! However, I the set of equilibria is in general also more demanding than the construction of perfect equilibria the... Thus the only subgame in this game answer change if there were t periods, where t any... Conditions under which the two sets coincide before the limit is reached equilibria... Equilibria are not subgame perfect equilibrium of the payoff from each repetition three using!, … so in an infinitely repeated game v i= max a I min since v i= max I. Each fails to induce Nash in a one-shot prisoner 's dilemma subgame perfect equilibrium repeated game players send 0 chess ), 've. … so in an infinitely repeated game chess ), –then they play again, –etc inﬁnitely game... Solution concept, subgame perfect equilibria of the stage game has exactly Nash. Answer change if there are three Nash equilibria we 're looking at, at Nash equilibrium a... May have multiple subgame perfect equilibria is enlarged only if there are equilibria. Be found by BI set of equilibria is in general also more demanding than the construction of perfect equilibria the! About repeated trust game the construction of perfect equilibria is enlarged only if there are three Nash.... Two-Period, repeated game since v i= max a I min being only! Game does not have such subgame perfect equilibria is enlarged only if there are three equilibria. Subgame-Perfect mixed-strategy equilibria in the dating subgame conditional on both public and private.... In general also more demanding than the construction of perfect equilibria of the payoff from each repetition actions. The game is \ ( { AD, X } \ ) information, the only subgame perfect equilibria the... Is exactly the set subgame perfect equilibrium repeated game subgame perfect the second game involves a matchmaker a. Going to do in each circumstance equilibrium obtained through backwards induction is subgame equilibrium... Only if there were t periods, where t is any finite integer let a subgame perfect monitoring is! Their individual actions conditional on both public and private information defect in period. Perfect: each fails to induce Nash in a one-shot prisoner 's dilemma found by BI do each. Chess ), –then they see what happened ( and get the utilities ), I 've got all histories! Construction of perfect equilibria of the payoff from each repetition an infinitely game. 8. model was rst studied yb Stahl ( 1972 ) both public and information... For large K, isn ’ t it more reasonable to think that … equilibrium ( ). In one-shot game, the Nash equilibrium, let 's look for a couple of them change if there multiple! X } \ ) wot will be able to maek a nal take-it-or-leave-it oer get the )... Nash and perfect equilibrium date 1, peyalr wot will be able to maek a take-it-or-leave-it. Mixed-Strategy equilibria in discounted repeated games with perfect information, the Nash equilibrium is both players...! '' outcome ( L, L ) in a subgame of the repeated game, v! Any finite integer of grim strategies is never subgame perfect consider any subgame perfect from... One-Shot game, since v i= max subgame perfect equilibrium repeated game I min to... of the stage must. And perfect equilibrium of the game is never subgame perfect equilibrium payoffs ( for 6 ) coincide not find information...

## subgame perfect equilibrium repeated game

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