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Adding information to one of the players does not increase the maxmin or the minmax value of the other players.
Suppose that a symmetric two-player game, in which each player has two pure strategies and all payoffs are nonnegative.
Explain why the evolutionarily stable strategy is that at which the number of male leopards born equals the number of females born.
Prove that if the payoff matrix of a two-player zero-sum game is antisymmetric, then the value of the game in mixed strategies is 0.
Is si chosen with positive probability in each of player i’s maxmin strategies? Prove this claim, or provide a counterexample.
Is si chosen with positive probability in one of player i’s maxmin strategies? Prove this claim, or provide a counterexample.
Find a mixed strategy of Player I that guarantees him the same payoff against any pure strategy of Player II.
What inequalities must the numbers a, b, c, d satisfy? Find the value in mixed strategies of this game.
Prove that in any n-person game, at Nash equilibrium, each player’s payoff is greater than or equal to his maxmin value.
For each of the following games, where Player I is the row player and Player II is the column player-Write out the mixed extension of the game.
The value of the two-player zero-sum game given by the matrix A is 0. Is it necessarily true that the value of the two-player zero-sum game.
Check whether there are other Nash equilibria in addition to those found by backward induction.
Show that the game has no value. (Each 0 here represents a matrix of the proper dimensions, such that all of its entries are 0.)
Show whether or not the value exists in each of the following games. If the value exists, find it and find all the optimal strategies for each player.
In a two-player zero-sum game on the unit square where Player I’s strategy set is X = [0, 1] and Player II’s strategy.
Consider a two-player non-zero-sum game on the unit square in which Player I’s strategy set is X = [0, 1].
Fifty people are playing the following game. Each player writes down, on a separate slip of paper, one integer in the set {0, 1,..., 100}, alongside his name.
Peter, Andrew, and James are playing the following game in which the winner is awarded M dollars.
Partnership Game Lee (Player 1), and Julie (Player 2), are business partners. Each of the partners has to determine the amount of effort.
Braess Paradox There are two main roads connecting San Francisco and San Jose, a northern road via Mountain View and a southern road via Cupertino.
The Davis Removal Company and its main rival, Roland Ltd, have fleets of ten trucks each, which leave the companies’ headquarters.
Location games Two competing coffee house chains, Pete’s Coffee and Caribou Coffee, are seeking locations for new branch stores in Cambridge.
Determine whether or not it can represent a strategic-form game corresponding to an extensive-form game with perfect information.
Let G be a game in extensive form. The agent-form game derived from G is a strategic-form game where each player i in G is split into several players.
Find a game that has at least one equilibrium, but in which iterative elimination of dominated strategies yields a game with no equilibria.