Stopping games are simple multi-player sequential games; each player has two actions: to continue or to stop. The game terminates once at least one player decides to stop. The terminal payoff depends on the moment at which the game stopped, on the subset of players who decided to stop at the terminal time, and on a state variable whose evolution is controlled by nature. In other words, the terminal payoff is some stochastic process. If no player ever stops, the payoff is 0 (this is without loss of generality).
Stopping games arise in various contexts: wars of attrition, duels, exiting from a shrinking market to name but a few. Does an equilibrium exist in such games? If the game is played in discrete time and payoffs are discounted, then payoffs are continuous over the strategy space, and an equilibrium exists. What happens if the game is played in continuous time?
Surprisingly, if there are at least three players, an equilibrium may fail to exist, even if the payoffs are constant (that is, they depend only on the subset of players who decide to stop, and not on the moment at which the game is stopped (and there is no state variable). Consider the game in the following figure:
In this game, in every time instant t, player 1 chooses a row, player 2 chooses a column, and player 3 chooses a matrix. Each entry except the (continue,continue,continue) entry corresponds to a situation in which at least one player stops, so that the three-dimensional vector in the entry is the terminal payoff in that situation. The sum of payoffs in every entry is 0, and therefore whatever the players play the sum of their payoffs is 0. Each player who stops alone receives 1, and therefore each one would like to be the first to stop. It is an exercise to verify that the game does not terminate at time t=0: termination at time 0 can happen only if there is a player who stops with probability 1 at time 0, but if, say, player 1 stops at time 0 then it is dominant for player 2 to continue at time t=0, and then it is dominant for player 3 to stop at time t=0, but then it is dominant for player 1 to continue at time t=0.
Thus, in this game, the players stop with some positive probability (that is smaller than 1) at time t=0, and, if the game has not terminated at time 0, each one tries to stop before the other. If the game is played in continuous time, there can be no equilibrium. Don’t worry because the payoff is not discounted; adding a discount factor will not affect the conclusion.
It is interesting to note that in discrete time this game has an equilibrium: in discrete time the players cannot fight about who stops first after time t=0, because the first time in which they can stop after time t=0 is time t=1, so they all stop with some positive probability at time t=1, and, in fact, they stop with positive probability at every discrete time t.
So stopping games are one example in which a game in continuous time does not have an equilibrium, while the corresponding game in discrete time does have an equilibrium.