1. Consider the variant of the collective decision-making described in class in
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1. Consider the variant of the collective decision-making described in class in which the chosen policy is the mean of the named policies. Does a player's strategy of naming his/her favorite policy weakly dominate the other strategies? 2. Two people have $10 to divide between themselves. Each person names a number (nonneg- ative integer) no more than 10. If the sum of the two numbers is at most 10, each person gets the named amount of dollars, and the rest of the money is destroyed. If the sum exceeds 10, and the numbers are different, the person who has named the smaller number, receives the corresponding number of dollars, and the second person receives the rest. If the sum exceeds 10, and the numbers are equal, each person receives $5. Determine the best response of each person to the other player's strategies, and find the Nash equilibria. 3. Two firms are developing a product for a market of fixed size. The longer the firm spends on the development, the better the product is. But the firm releasing the product first has an advantage: a customer may not switch to a better product once got used to a worse one. We model the situation in the following way. A firm that releases first at time t gets h(t) share of the market, leaving the remainder to the competitor. We assume that h(t) is an increasing function of t with h(0) = 0 and h(T) = 1 . If the firms release simultaneously, each gets half of the market. Represent this situation as a strategic game and find its Nash equilibria 4. Players 1 and 2 each choose a member of the set f1,... ,K). If the players choose the $1 to the player 1: otherwise no payment is made. Each player same number then player 2 maximizes his expected monetary payoff. Find the mixed strategy Nash equilibria of this game pays 5. In the Cournot's duopoly game with linear demand function and costs, suppose the unif costs of the two firms are different (assume 2Explanation / Answer
1)Yes, it can be claimed that for each player i, the action of naming her favourite policy xi weakly dominates all her other actions. The reason is that relative to the situation in which she names xi, she can change the median only by naming a policy further from her favourite policy than the current median; no change in the policy she names moves the median closer to her favourite policy.
So ,variant of the mechanism for collective decision-making in which the policy chosen is the mean, rather than the median, of the policies named by the players and this player's action of naming her favorite policy weakly dominate all her other actions
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