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Quitting Games

Quitting Games. Eilon Solan , Tel Aviv University. SING 15, Turku; July 3, 2019. Stopping Games: Model. There are n players. In every stage each player decides whether to stop the game or to continue . If at least one player stops , the game terminates.

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Quitting Games

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  1. Quitting Games EilonSolan, Tel Aviv University SING 15, Turku; July 3, 2019

  2. Stopping Games: Model • There are n players. • In every stage each player decides whether to stop the game or to continue. • If at least one player stops, the game terminates. • The terminal payoff depends on the set of players who stop at the stopping stage, on the stage, and on a state variable, which is constant over time. • If no player stops in the current stage, the players get some information on the state variable. • If no player ever stops, the payoff is 0.

  3. Stopping Games: Questions • When to stop? • How does the equilibrium change when the parameters change? A More Basic Question • Does an ε-equilibrium exist? A Simple Class of Games Quitting games = stopping games with deterministic constant payoffs (independent of time).

  4. Existence of ɛ-Equilibrium in Randomized Stopping Times Discrete Time Continuous Time Two-player zero-sum Rosenberg, Solan, and Vieille (2001) Laraki and Solan (2005) Two-player nonzero-sum Shmaya and Solan (2004) Laraki and Solan (2013) Solan (1999) + Shmaya and Solan (2004) Counterexample: Laraki, Solan, and Vieille (2005) Three players More than three players Open Problem Correlated Eq. Heller (2012) Sunspot Eq. Solan and Solan (2019)

  5. 0,1,3 1,3,0 Stop Cont Cont Stop Cont Stop Cont 3,0,1 1,1,0 Stop 1,0,1 0,1,1 0,0,0 Three-Player Discrete Time Quitting game • Example (Flesch, Thuijsman, Vrieze, 1997) • If nobody ever Stops, the payoff is (0,0,0). Janos Flesch Frank Thuijsman KoosVrieze Player 3: Player 2: Player 1:

  6. 0,1,3 1,3,0 Stop Cont Cont Stop Cont Stop Cont 3,0,1 1,1,0 Stop 1,0,1 0,1,1 0,0,0 Three-Player Discrete Time Stage 1: Play (½ C + ½ S, C, C). Stage 2: Play (C, ½ C + ½ S, C). Stage 3: Play (C, C, ½ C + ½ S). Continue cyclically. Janos Flesch Frank Thuijsman KoosVrieze Expected payoff: g = (1,2,1). Player 3: Player 2: Player 1:

  7. 0,1,3 1,3,0 Stop Cont Cont Stop Cont Stop Cont 3,0,1 1,1,0 Stop 1,0,1 0,1,1 0,0,0 Three-Player Discrete Time Stage 1: Play (½ C + ½ S, C, C). Stage 2: Play (C, ½ C + ½ S, C). Stage 3: Play (C, C, ½ C + ½ S). Continue cyclically. Expected payoff: g = (1,2,1). Expected payoff from 2nd stage: (1,1,2). Janos Flesch Frank Thuijsman KoosVrieze Player 3: Player 2: 1,1,2 Player 1:

  8. 0,1,3 1,3,0 Stop Cont Cont Stop Cont Stop Cont 3,0,1 1,1,0 Stop 1,0,1 0,1,1 0,0,0 Three-Player Discrete Time Stage 1: Play (½ C + ½ S, C, C). Stage 2: Play (C, ½ C + ½ S, C). Stage 3: Play (C, C, ½ C + ½ S). Continue cyclically. Janos Flesch Frank Thuijsman KoosVrieze (½ C + ½S, C, C) is an equilibrium of this one-shot game. Player 3: Player 2: 1,1,2 Player 1:

  9. 0,1,3 1,3,0 Stop Cont Cont Stop Cont Stop Cont 3,0,1 1,1,0 Stop 1,0,1 0,1,1 0,0,0 Three-Player Discrete Time For every x=(x1,x2,x3), consider the one-shot game G(x) where the payoff if everyone continues is x. E(x) = the set of equilibrium payoffs in this game. (1,1,2) is in E(1,2,1) (2,1,1) is in E(1,1,2) (1,2,1)is in E(2,1,1) (1,2,1) ; (1,1,2) ; (2,1,1) is a periodic orbit of E. Player 3: Player 2: x1,x2,x3 Player 1:

  10. 0,1,3 1,3,0 Stop Cont Cont Stop Cont Stop Cont 3,0,1 1,1,0 Stop 1,0,1 0,1,1 0,0,0 Three-Player Discrete Time For every x=(x1,x2,x3), consider the one-shot game G(x) where the payoff if everyone continues is x. E(x) = the set of equilibrium payoffs in this game. (1,2,1) ; (1,1,2) ; (2,1,1) is a terminating periodic orbit of E. (1.5,1.5,1.5) is a non-terminating fixed point of E. Player 3: Player 2: x1,x2,x3 Player 1:

  11. 0,1,3 1,3,0 Stop Cont Cont Stop Cont Stop Cont 3,0,1 1,1,0 Stop 1,0,1 0,1,1 0,0,0 Three-Player Discrete Time Question: Does the set-valued function E always have a terminating periodic orbit? Theorem (Solan and Vieille, 2001): If (a) the payoff to a player when he stops alone is 1, and (b) the payoff to a player when he stops with others is at most 1, then (a perturbation of) E has a terminating periodic orbit, which is an ε-equilibrium. Player 3: Player 2: x1,x2,x3 Player 1:

  12. Correlated Equilibrium Before the game starts, a mediator chooses a vector of messages (m1, m2, …, mn) for the n players according to a correlated distribution, and privately sends to each player the message chosen for her. An ε-equilibrium in the game with mediator is a normal-form correlated ε-equilibrium. Theorem (Solan and Vohra, 2001; Heller, 2012): Every stopping game admits a normal-form correlated ε-equilibrium. Yuval Heller RakeshVohra

  13. 0,1,3 1,3,0 Stop Cont Cont Stop Cont Stop Cont 3,0,1 1,1,0 Stop 1,0,1 0,1,1 0,0,0 Correlated Equilibrium: Example In the example: the mediator selects a player, each player with probability 1/3, tells the chosen player to stop, and the other two players to continue. Player 3: Player 2: Player 1:

  14. Sunspot Equilibrium At every stage t, the players observe a random signal, which is uniformly distributed in [0,1], independent of past play. An ε-equilibrium in the game with public signals is a sunspotε-equilibrium of the original game. Theorem (Solan and Solan, 2019): Every stopping game admits a sunspot ε-equilibrium. Omri Solan

  15. Example Payoff if Player 1 stops alone = (0,4,-1,-1) Payoff if Player 2 stops alone = (4,0,-1,-1) Payoff if Player 3 stops alone = (-1,-1,0,4) Payoff if Player 4 stops alone = (-1,-1,4,0) We will implement (0,0,1,1) and (1,1,0,0) as a sunspot ɛ-equilibrium payoff. g(0,0,1,1) = ½ g(0,0,0,2) + ½ g(0,0,2,0) g(0,0,0,2) = ½ (-1,-1,0,4) + ½ g(1,1,0,0) g(0,0,2,0) = ½ (-1,-1,4,0) + ½ g(1,1,0,0) (0,0,1,1) = ½ (0,0,0,2) + ½ (0,0,2,0) (0,0,0,2) = ½ (-1,-1,0,4) + ½ (1,1,0,0) (0,0,2,0) = ½ (-1,-1,4,0) + ½ (1,1,0,0) • Nature chooses whether we implement (0,0,0,2) or (0,0,2,0). • If we implement (0,0,0,2), Player 3 stops with probability 1/2. • If we implement (0,0,2,0), Player 4 stops with probability 1/2. • If the designated player did not stops, from the following stage and on we will implement (1,1,0,0).

  16. Continuous Time: Counterexample Continue Stop Continue Stop Continue Stop Continue -1, 1, 0 0, -1, 1 1, 0, -1 Stop 1, 0, -1 0, -1, 1 -1, 1, 0 0, 0, 0 Features of the game: 1) Sum of payoffs is 0. Payoffs are cyclic. 2) If one player stops alone, he receives 1. Suppose there is an equilibrium. Case 1: the game stops with probability 1 at time 0. W.l.o.g. Player 1 Stops at time 0 with probability 1. Nicolas Vieille Rida Laraki

  17. Continuous Time: Counterexample Continue Stop Continue Stop Continue Stop Continue -1, 1, 0 0, -1, 1 1, 0, -1 Stop 1, 0, -1 0, -1, 1 -1, 1, 0 0, 0, 0 Features of the game: 1) Sum of payoffs is 0. Payoffs are cyclic. 2) If one player stops alone, he receives 1. Suppose there is an equilibrium. Case 1: the game stops with probability 1 at time 0. W.l.o.g. Player 1 Stops at time 0 with probability 1. Continue is dominating for Player 2. Nicolas Vieille Rida Laraki

  18. Continuous Time: Counterexample Continue Stop Continue Stop Continue Stop Continue -1, 1, 0 0, -1, 1 1, 0, -1 Stop 1, 0, -1 0, -1, 1 -1, 1, 0 0, 0, 0 Features of the game: 1) Sum of payoffs is 0. Payoffs are cyclic. 2) If one player stops alone, he receives 1. Suppose there is an equilibrium. Case 1: the game stops with probability 1 at time 0. W.l.o.g. Player 1 Stops at time 0 with probability 1. Continue is dominating for Player 2. Stop is dominating for Player 3. Nicolas Vieille Rida Laraki

  19. Continuous Time: Counterexample Continue Stop Continue Stop Continue Stop Continue -1, 1, 0 0, -1, 1 1, 0, -1 Stop 1, 0, -1 0, -1, 1 -1, 1, 0 0, 0, 0 Features of the game: 1) Sum of payoffs is 0. Payoffs are cyclic. 2) If one player stops alone, he receives 1. Suppose there is an equilibrium. Case 1: the game stops with probability 1 at time 0. W.l.o.g. Player 1 Stops at time 0 with probability 1. Continue is dominating for Player 2. Stop is dominating for Player 3. Continue is dominating for Player 1. Nicolas Vieille Rida Laraki

  20. Continuous Time: Counterexample Continue Stop Continue Stop Continue Stop Continue -1, 1, 0 0, -1, 1 1, 0, -1 Stop 1, 0, -1 0, -1, 1 -1, 1, 0 0, 0, 0 Features of the game: 1) Sum of payoffs is 0. Payoffs are cyclic. 2) If one player stops alone, he receives 1. Case 2: the game continues with positive probability after time 0. By Feature 1, the continuation payoff of at least one of the players is non-positive. He wants to “stop first”.

  21. Summary • Stopping games are a class of dynamic games that are useful in applications. • Existence of ε-equilibrium when at least four players are involved is an open problem. • Normal-form correlated ε-equilibrium and sunspot ε-equilibrium always exists. • Computation: normal-form  , sunspot  • Characterization: normal-form  , sunspot  • How do they change when the parameters of the game change?

  22. شكرا dziękuję Merci Hvala Kiitos 謝謝 תודה רבה Dank ありがとう Thank you Danke eskerrikasko спасибо Köszönöm

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