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1 Perfect Correlated Equilibria in Stopping Games Yuval Heller Tel-Aviv University (Part of my Ph.D. thesis supervised by Eilon Solan) http://www.tau.ac.il/~helleryu/ 3 rd Israeli Game Theory Conference December 2008

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Introduction: Stopping games perfect correlated ( )-equilibrium Main Result Summary 2 Proof Outline Reductions Finite trees & absorbing games Stochastic variation of Ramsey theorem Equilibrium construction

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3 Stopping Games (Undiscounted, Multi-player, Discrete time) Finite set of players: I Unknown state variable: (state space) Filtration: F =( F n ) At each stage n the players receive a symmetric partial information about the state : F n ( )

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4 Stopping Games (undiscounted, multi-player, discrete time) Stage 1 - everyone is active Stage n: All active players simultaneously declare whether they stop or continue A player that stops become passive for the rest of the game Player’s payoff depends on the history of players’ actions while he has been active and on the state variable

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5 Literature: 2-player zero-sum Stopping Games Dynkin (1969) – introduction, value where simultaneous stops are not allowed Neveu (1975) – value when each player prefers the other to stop Rosenberg, Solan & Vieille (2001) – use of randomized strategies, value with payoffs’ integrability

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6 Literature: 2-player non-zerosum Stopping Games Existence of approximate Nash equilibrium when the payoffs have a special structure: Morimoto (86), Mamer (87), Ohtsubo (87, 91), Nowak & Szajowski (99), Neumann, Ramsey & Szajowski (02) Recently, Shmaya & Solan (04) proved existence assuming only integrability Multi-player stopping games: no existence results

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7 Stopping Games - Applications Most applications in the literature: Payoffs: Specific assumptions, such as monotony Discount factor 2 players Multi-player variations are natural

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8 Struggle of survival in a declining market At each turn, each firm loses money A firm can stay or exit the market for good Partial production is inefficient Market is more profitable with less firms Which firms survive? What is the exit order? Ghemawat & Nalebuff (1985)... Steel market in 70’s and 80’s

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9 Research & Development Race for developing a patent At each turn, continue spending money on research or leave the race The first firm to complete the patent earns a lot Stochastic function of spent money Fudenberg & Tirole (1985)…

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10 War of attrition Attrition wars among animals: Becoming the leader (alpha-male) Territory Maynard-Smith (1982), Nalebuff & Riley (1985)… 2 nd price auctions where all bidders pay Krishna & Morgan (1997)…. Political Sciences – lobbying Bulow & Klemperer (2001)

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Introduction: Stopping games perfect correlated ( )-equilibrium Main Result Summary 11 Proof Outline Reductions Finite trees & absorbing games Stochastic variation of Ramsey theorem Equilibrium construction

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12 Perfect Equilibrium Nash equilibrium may be sustained by non-credible threats of punishment Punisher receives a low payoff The stronger concept of perfect equilibrium (Selten, 1965, 1975) has been studied. Examples: Fine & Li (1989): uniqueness in discounted 2-player games with monotone payoffs Mashiah-Yaakovi (2008) – existence of ( )-perfect equilibrium when simultaneous stops aren’t allowed

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13 Correlated Equilibrium Aumann (1974): An equilibrium in an extended game with a correlation device Device D sends each player i a private signal m i M i (M= i M i ) before the game starts according to (M) The extended game G(D) Consistent with Bayesian decision making (Aumann, 87) Other appealing properties: computability, linear equations, closed and convex set

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14 Correlated Equilibrium in Sequential Games Two main versions: Normal-form: signals are sent only before the game starts Extensive-form: signals are sent at each stage Equilibrium: normal-form extensive-form Correlation among players is natural in many setups: Countries negotiate actions Firms choose strategies based on market’s history A manager coordinates the actions of his workers

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15 Normal-Form Correlation (1) Sometimes players may coordinate before play starts but coordination along the play is costly / impossible: Example (1) - war of attrition in nature: Commonly modeled as stopping games Coordination before play starts is implemented by evolution of phenotype roles E.g.: Shmida & Peleg, 1997

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16 Normal-Form Correlation (2) Example (2) - News playing among day traders: Monthly employment report will be published at noon Several minutes elapse before market adjusts New information gradually arrives during that time Quick trading can be profitable See e.g., Christie-David, Chaudhry & Khan (2002) Traders of a firm can coordinate their actions in advance Coordination along the play is costly (time limit) Traders may have different payoffs

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17 ( )-Perfect Correlated Equilibrium – A bound for the probability of: An event E Correlation device sends a signal in M’ M >0 – A bound for the maximal profit a player can earn by deviating at any stage and after any history, conditioned on that E and m M’ Extending the definitions for finite games: Myerson (1986), Dhillon & Mertens (1996)

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Introduction: Stopping games perfect correlated ( )-equilibrium Main Result Summary 18 Proof Outline Reductions Finite trees & absorbing games Stochastic variation of Ramsey theorem Equilibrium construction

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19 Main Result For every >0, a multi-player stopping game admits a normal-form uniform perfect correlated ( )-equilibrium with a universal correlation device Uniform: An approximate equilibrium in any long enough finite game and in any discounted game with high enough discount factor Universal device – doesn’t depend on game payoffs Corollary: Uniform perfect correlated equilibrium payoff

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Introduction: Stopping games perfect correlated ( )-equilibrium Main Result Proof Outline Reductions Finite trees & absorbing games Stochastic variation of Ramsey theorem Equilibrium construction 20 Summary

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1.Terminating games: game terminates at the first stop 2.Tree-like games (Shmaya & Solan, 03) : for every n, F n is finite A finite collection of matrix payoffs 3.Deep enough in the tree: with high probability any matrix payoff either: Repeats infinitely often Never occurs 21 Reductions

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22 Reductions Reductions require 2 properties from the equilibrium ( , -unrevealing - expected payoff of each player “almost” doesn’t change With probability of at-least 1- , changes by less than Universal - The correlation device D(G, , ) depends only on |I| and D(G, , )=D(|I|, )

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Introduction: Stopping games perfect correlated ( )-equilibrium Main Result Proof Outline Finite trees & absorbing games Stochastic variation of Ramsey theorem Equilibrium construction 23 Summary Reductions

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24 Games on Finite Trees Equivalent to an absorbing game: A stochastic game with a single non-absorbing state. 2 special properties: Recursive game – Payoff in non-absorbing states is 0 Single non-absorbing action profile

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25 Games on Finite Trees An adaptation of a result of Solan & Vohra (2002): A game on a finite tree has one of the following: 1.Non-absorbing equilibrium (game never stops) 2.Stationary absorbing equilibrium. Adaptations: Perfection Limit minimal per-round terminating probability 3.A special distribution: allows to construct a correlated -equilibrium. Adaptations: unrevealing, universal device

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Introduction: Stopping games perfect correlated ( )-equilibrium Main Result Proof Outline Finite trees & absorbing games Stochastic variation of Ramsey theorem Equilibrium construction 26 Summary Reductions

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27 Ramsey Theorem (1930) A finite set of colors Each 2 integers (k,n) are colored by c(k,n) There is an infinite sequence of integers k 1 <k 2 <k 3 <… such that: c(k 1,k 2 ) =c(k i,k j ) for all i<j 0123456789101112 k1k1

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28 Stochastic Variation of Ramsey Theorem (Shmaya & Solan, 04) Coloring each finite sub-tree. There is an increasing sequence of stopping times: 1 1- Low probability 11

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Introduction: Stopping games perfect correlated ( )-equilibrium Main Result Proof Outline Finite trees & absorbing games Stochastic variation of Ramsey theorem Equilibrium construction 29 Summary Reductions

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30 Equilibrium Construction Each finite tree is colored according to: Whether it has a non-absorbing perfect equilibrium, an absorbing perfect equilibrium, or a special distribution The equilibrium payoff The maximal payoffs when a player stops alone If c implies that each game on finite tree has a perfect equilibrium, concatenate the equilibria to obtain an approximate perfect equilibrium of G

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31 Equilibrium Construction Last case: c implies that a special distribution exists This allow to construct an approximate unrevealing perfect correlated equilibrium with a universal correlation device An adaptation of the protocol of Solan and Vohra (01)

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Introduction: Stopping games perfect correlated ( )-equilibrium Main Result Proof Outline Finite trees & absorbing games Stochastic variation of Ramsey theorem Equilibrium construction 32 Summary Reductions

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33 Summary and Future Research Summary: every multi-player stopping game admits an approximate normal-form uniform perfect correlated equilibrium with a universal correlation device Future research: Using this notion of equilibrium in the study of other dynamic games Structure of uniform perfect correlated equilibrium payoffs in specific applications

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34 Questions & Comments? Y. Heller (2008), Perfect correlated equilibria in stopping games, mimeo. http://www.tau.ac.il/~helleryu/

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Introduction: Stopping games perfect correlated ( )-equilibrium Main Result Proof Outline Finite trees & absorbing games Stochastic variation of Ramsey theorem Equilibrium construction 35 Summary Reductions

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36 Reduction to Terminating Games Proposition: Every game that stops immediately admits a ( , )- unrevealing perfect correlated ( )-equilibrium with a universal correlation device Every stopping game admits the same kind of ( )-equilibrium

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37 Proof Outline Induction on the number of players Given a stopping game G, we define an auxiliary terminating game G’: The payoff to I \ S when a coalition S stops is the equilibrium payoff in the induced stopping game G’ admits an unrevealing perfect correlated ( )-equilibrium with a universal correlation device Concatenation gives such an equilibrium in G

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38 Tree-like Games Shmaya & Solan (2002) showed that any stopping game can be approximated by a tree-like stopping game, with the same set of approximate equilibria Small perturbations of the payoffs don’t change the set of approximate equilibria we can assume that the payoff process has a finite range Each set F n F n can be identified with a node in a tree

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Tree-like Games – Shmaya & Solan’s Proof Outline k th partition: Discretization of the game: Depth: k Precision: Refinement of all previous partitions Defines the k th approximating game on a tree The game on finite tree that begins on m and ends on l will be played on the m+l approximating game

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40 Tree-like Games F4F4F4F4 F1F1 F2F2 F4F4

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41 Deep Enough in the Tree FnFn v1v1 v1v1 v1v1 v1v1 v 1, v 3, v 5 occur infinitely often, all other v V do not occur at all v1v1 v1v1 v1v1 G (F n ): The induced game that begins at the node F n

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42 Lemma - Induced Games Let: G - a terminating game, - a stopping time Every induced game G (F n ), where F n is in the range of , admits an unrevealing perfect correlated ( )- equilibrium with a universal correlation device G admits the same kind of (C· )-equilibrium Corollary: We can assume to be “deep enough”

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Proof Outline Until the players follow an equilibrium in a finite stopping game with absorbing states {F } with payoffs {x F equilibria payoffs of G (F ) After players follow ( )-equilibrium of G (F ) Relying on that the equilibrium is unrevealing and with a universal correlation device Illustration…. 43

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Proof Outline FF 44 x1x1 x2x2 x3x3 V(|I|, ) - universal correlation device x4x4 x5x5 x6x6 x7x7

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45 Games on Finite Trees g i : maximal payoff player i can get by stopping alone The special distribution over (nodes · players): A stopping player i and a node with maximal payoff (R i i,n =g i ) The distribution gives each player i at-least g i Each stopping player has a punisher j that stops when R j i,n <g i Allows to construct a correlated -equilibrium

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46 Stationary Absorbing Equilibrium: Adaptations Perfection - using a perturbed tree with probability to ignore players’ requests to stop Limiting the minimal per-round terminating probability (adapting the methods of Shmaya & Solan, 2004) If there is a player i with a payoff below g i, then can’t be too small or player i stops when his payoff is g i Otherwise either case 3 applies, or there is a node where at-least 2 players stop with a non-negligible probability Recursive trimming of such nodes gives the needed limit

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47 Last case: c implies that a special distribution exists Let i k be the k-th time that player i’s maximal payoff occur with the requirement i k > j k-1 for all i, j Using the fact that we are “deep enough” in the tree An approximate unrevealing perfect correlated equilibrium with a universal correlation device is constructed as follows… Equilibrium Construction: Protocol Description `

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48 Equilibrium Construction: Protocol Description A quitter i’ is secretly chosen according to the special distribution A number l’ is chosen uniformly in {1,T’} i’ receives the signal l’ A number l is chosen uniformly in {l’+1,l’+T} 1<<T<<T‘ The punisher of i’ receives the signal l Each other player receives a signal l+1 Approximate unrevealing perfect correlated equilibrium: each player stops at l (when l is his signal) modulo 1+T+T’

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Introduction: Stopping games perfect correlated ( )-equilibrium Main Result Proof Outline Finite trees & absorbing games Stochastic variation of Ramsey theorem Equilibrium construction 49 Summary Reductions

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