1. Eilon Solan, Tel Aviv University
Quitting Games
Eilon Solan, 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. 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
Koos Vrieze
1,3,0
0,1,3
3,0,1
0,1,1
1,1,0
0,0,0
1,0,1
Cont
Stop
Player 1:
Player 2:
Player 3:

6. 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
Koos Vrieze
Expected payoff: g = (1,2,1).
1,3,0
0,1,3
3,0,1
0,1,1
1,1,0
0,0,0
1,0,1
Cont
Stop
Player 1:
Player 2:
Player 3:

7. 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
Koos Vrieze
1,3,0
0,1,3
3,0,1
0,1,1
1,1,0
0,0,0
1,0,1
Cont
Stop
Player 1:
Player 2:
Player 3:
1,1,2

8. 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
Koos Vrieze
(½ C + ½ S, C, C) is an equilibrium of this one-shot game.
1,3,0
0,1,3
3,0,1
0,1,1
1,1,0
0,0,0
1,0,1
Cont
Stop
Player 1:
Player 2:
Player 3:
1,1,2

9. 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.
1,3,0
0,1,3
3,0,1
0,1,1
1,1,0
0,0,0
1,0,1
Cont
Stop
Player 1:
Player 2:
Player 3:
x1,x2,x3

10. 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.
1,3,0
0,1,3
3,0,1
0,1,1
1,1,0
0,0,0
1,0,1
Cont
Stop
Player 1:
Player 2:
Player 3:
x1,x2,x3

11. 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.
1,3,0
0,1,3
3,0,1
0,1,1
1,1,0
0,0,0
1,0,1
Cont
Stop
Player 1:
Player 2:
Player 3:
x1,x2,x3

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.
Payoff to player i is E[ XS*τ* 1(τ* < ∞) ] .
Theorem (Solan and Vohra, 2001; Heller, 2012): Every stopping game admits a normal-form correlated ε-equilibrium.
Yuval
Heller
Rakesh Vohra

13. 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.
1,3,0
0,1,3
3,0,1
0,1,1
1,1,0
0,0,0
1,0,1
Cont
Stop
Player 1:
Player 2:
Player 3:

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 sunspot ε-equilibrium of the original game.
Payoff to player i is E[ XS*τ* 1(τ* < ∞) ] .
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)
We will implement (0,0,1,1) and (1,1,0,0) as a sunspot ɛ-equilibrium payoff.
(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)
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)
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
Stop
Continue
-1, 1, 0
0, -1, 1
1, 0, -1
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
Stop
Continue
-1, 1, 0
0, -1, 1
1, 0, -1
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
Stop
Continue
-1, 1, 0
0, -1, 1
1, 0, -1
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
Stop
Continue
-1, 1, 0
0, -1, 1
1, 0, -1
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
Stop
Continue
-1, 1, 0
0, -1, 1
1, 0, -1
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?

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