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Players i, j are symmetric players if for every coalition S that does not include any one of them,v(S ∪ {i}) = v(S ∪ {j}).
(a) Prove that the symmetry relation between two players is transitive: if i and j are symmetric players, and j and k are symmetric players, then i and k are symmetric players.
(b) Show that if the core is nonempty, then there exists an imputation x in the core that grants every pair of symmetric players the same payoff, i.e., xi = xj for every pair of symmetric players i, j .
You randomly call friends who could be potential partners for a dance. You think that they all respond to your requests independently of each other, and you estimate that each one s 7% likely to accept your request. Let X denote the number of call..
A 5-card poker hand is said to be a full house if it consists of 3 cards of the same denomination and two other cards of the same denomination(of course,different from the first denomination). How many hands of full house are possible?
Is there any NE in whicheveryone votes? Is there any NE in which there is a tie and not everyone votes? Is there any NE in which one of the candidates wins by one vote?
Describe this situation as a strategic-form game, in which each driver chooses the route he will take.- What are all the Nash equilibria of this game? At these equilibria, how much time does the trip take at an early morning hour?
Prove the following claims:- If strategy σk approaches a set C for player k, then it approaches every supersetof C for that player.
Suppose your friend is creating a game about Lewis and Clark's expedition. Provide a list of 10 possible elements she should include in the game in order to make it as informative and exciting as possible
Write down the base game for this situation.- Find all the equilibria of the one-stage game (the base game).- Find all the equilibria of the two-stage game.
Find all subgame perfect equilibria of this game. Now suppose player 2 has found a way of cheating, getting to observe player 1's hand. Represent the extensive form of this game and find its subgame perfect equilibria.
Model this situation as a Bayesian game and show that in any Nash equilibrium the highest prize that either individual is willing to exchange is the smallest possible prize.
Show that army 2 can increase its subgame perfect equilibrium payoff (and reduce army 1's payoff) by burning the bridge to its mainland, eliminating its option to retreat if attacked.
After researching cysts of a particular type, a doctor learns that out of 10,000 such cysts examined, 1,500 are malignant and 8,500 are benign. A diagnostic test is available which is accurate 80% of the time (whether the cyst is malignant or not)..
Find all the pure-strategy subgame-perfect equilibria with no discounting (δ = 1). Be precise in defining history-contingent strategies for both players.
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