Game Theory True/False

The SPE in the trust game captures mutual trust between players.

False

In Stackelberg game, both players' equilibrium payoffs are strictly lower than that in Cournot game since the follower can respond to the leader's actual choice.

False, It is true only for the follower. The leader's payoff is strictly higher.

Some SPE is not a Nash equilibrium.

False, SPE is a refinement of Nash equilibrium.

If there are two players, and player 1 has a weakly dominated strategy, player 1 never plays the strategy in Nash equilibrium

False, a weakly dominated strategy can still be the best response and therefore played in a nash equilibrium

Any extensive form game has more than one subgame.

False, consider the simultaneous BoS which only has one subgame.

Whenever a game has a pure strategy Nash equilibrium, the game has no mixed strategy Nash equilibrium.

False, e.g. BoS has one mixed strategy Nash even though it has two pure strategy Nash

If (A1*, A2*) is a Nash equilibrium, then U2(A1*, A2*) ≥ U2(A1, A2) for all (A1, A2)

False, e.g. in the case of two equilibria another nash equilibrium may have a higher utility.

Whenever a game has a Nash equilibrium, there are more than one Nash equilibrium.

False, e.g. prisoner's dilemma only has one NE

There is a sequence of 100 domino and 100th is blue. Suppose we know that if n-th domino is blue, n−2-th domino is also blue. Can we conclude all the domino are blue by mathematical induction?

False, if 99th is red but all others are blue, the law is still satisfied.

In stag hunt game, the equilibrium with "stag," which requires successful coordination, is the risk dominant equilibrium, whereas the equilibrium with "rabbit," which guarantees the secure payoff is the payoff dominant equilibrium.

False, its the other way around

In any mixed strategy equilibrium, all pure strategies are indifferent for each player

False, some strategies might be strictly dominated and have a probability of 0

SPE eliminates Nash equilibria that rely on non-credible threat

True

There can be a Nash equilibrium in which one player selects a mixed strategy, whereas the other player selects a pure strategy.

True, a probability of 1 can be mixed with other probabilities

Suppose player 2, the first mover, chooses an action from {1, 2, 3, 4} and then player 1 chooses an action from {L, R}. If player 1 cannot observe player 2's action, then player 1's information set consists of four decision nodes.

True, as player 1 cannot observe the four decision nodes the information set consists of 4 decision nodes.

There can be more than one SPE in some games.

True, e.g. consider a co-ordination game, each SPE has the same payoff.

In any sequential game with finite periods, if there is a strict dominant strategy for every player, the Nash equilibrium and the SPE are the same.

True, each plays the strictly dominant strategy

If A1 is never a best response to any action, there is no Nash equilibrium in which player 1 chooses A1.

True, if A1 cannot be a best response, it won't be played in an equilibrium