Two players, both with zero wealth, bargain over how to divide £X > 0 between them. Failure to reach agreement means both get nothing. Both players are expected utility max- imisers. Player 1 has utility u(x) = x“, where 0 < a < 1. Player 2 has utility u(x) = where 0 < B < 1. Determine the Nash solution for this problem
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- Find a separating perfect Bayes–Nash equilibrium for the figure below.Here is a Bayesian BOS game. Man has a special preference t1 on boxing while woman has a special preference t2 on opera. Both are in [0,2]. The preference information is private. i.e. Each knows his (her) own preference but does not know the other's preference. The below is the payoff. M/W Boxing Opera Воxing 3+t1,1 0,0 Opera 0,0 1,3+t2 We are checking whether the below strategy profile is a Bayesian Nash Equilibrium. "Man chooses boxing if t1>p while woman chooses opera if t2>g." 1) Given t1, man will choose boxing if t1> /g- 2) Given t2, woman will choose opera if t2> /p- 4) By shoving the simultaneous equations, we get p=q=sqrt(6)-Apple and Google are interested in hiring a new CEO. Both firms have the same set of final candidates for the CEO position: Indra, Cao, and Virginia. Both firms need to decide who to make a job offer to, and the hiring process is such that they each only make one job offer. If, say, Apple makes a job offer to Indra and Google makes a job offer to one of the other candidates, then Apple’s probability of success in hiring Indra is pIndra. The same is true for Google. If they both make a job offer to Indra, each has probability pIndra/2 of success. It has been estimated that pIndra = 20%, and pCao = pVirginia = 30% (Note that these probabilities need not add up to 100%). Suppose that both Apple and Google attach a valuation of 10 to successfully hiring Indra, and a valuation of 7 to successfully hiring each of the other candidates. A hiring attempt, if unsuccessful, has a valuation of zero. Convert this story into a game by completing the following game table; Google…
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- Suppose that, in an ultimatum game, the proposer may not propose less than $1 nor fractional amounts, and therefore must propose $1, $2, …, or $10 (see image attached below). The responder must Accept (A) or Reject (R). Suppose, first, that this game is played by two egoists, for whom u(x,y)=x. Find all subgame-perfect equilibria in this game. Suppose, second, that this game is played by two altruists, for whom u(x,y) = ⅔(x)1/2 + ⅓(y)1/2. Find all subgame-perfect equilibria in this game.3. Two individuals are involved in a synergistic relationship. If both individuals devote more effort to the relationship, they are both better off. Specifically, an effort level is a nonnegative number, and player 1's payoff function is e₁(1+€₂-e₁), where ei is player i's effort level. For player 2 the cost of effort is either the same as that of player 1, and hence her payoff function is given by e2(1+e₁ - e₂), or effort is very costly for her in which case her payoff function is given by e₂(1 + e₁ — 2e2). 1 Player 2 knows player 1's payoff function and whether the cost of effort is high for herself or not. Player 1, however, is uncertain about player 2's cost of effort. He believes that the cost of effort is low with probability p, and high with probability 1 - p, where 0 < p < 1. Find the Bayesian Nash equilibrium of this game as a function of p.Olivia is thinking about opening a new bakery (the entrant). There is already a bakery open in her neighborhood (the incumbent), and the owner of the incumbent bakery makes it clear that if Olivia enters the market, they will cut their prices in an attempt to drive the new bakery out of business. Based on the payoff matrix below, is the incumbent’s threat credible? That is, if Olivia opens a new bakery, will the incumbent actually lower their prices? Note: the entrant chooses the row, the incumbent chooses the column High price Low price Enter 1, 2 -1, 1 Don’t enter 0, 10 0, 1 a. Yes, the threat is credible b. No, the threat is not credible
- Consider the following three-stage duopoly game. There are two firms in the market: Firm 1 and Firm 2. In the first stage, a first-price auction is conducted to determine the order of moves. The firm with the highest bid (first mover) pays a cost equivalent to its bid and chooses its action in the second stage. After observing this action, the other firm (second mover) chooses its action in the third stage. When there is a tie in the first stage (both firms submit the same bid), a coin is tossed to determine the winner, and only the winner pays its bid. Each firm has the following three possible actions: small (expansion), medium (expansion) and large (expansion), and the payoffs they obtain in the market (not the final payoffs yet) are shown below: The second mover Small Medium Large 16, 12 Medium 18, 8 19, 7 13, 10 14, 11 Small 9, 9 10, 13 12, 10 The first mover Large 14, 12 The final payoff is equivalent to the payoff obtained in the market minus the cost (if any) paid in auction.…In the normal form game below, the payoff matrix depends on the parameter a. 1 1 2 ABC a 0,2 6,4 -2,4 b 0, -24 -2a, 2a 8, 2 C 4,0 0,4 0, 14 Find the values of a, for which at least one pure strategy Nash equilibrium exists. Compute the value of a for which the expected payoff is the same for both players when the mixed strategy profile (01,02) = ((1/3,2/3,0), (0,1/4,3/4)) is played. Find the best response of player 1 (as a function of the parameter a) to player's 2 mixed strategy o2 = (1/2,1/2,0). Assuming a = 0 eliminate iteratively all dominated strategies and find a mixed strategy Nash equilibrium in this game.Problem 4: Consider an infinitely repeated game, where the base game is the following 2-person 2x2 game: A A 0,0 10, 10 S1: choose A always S2: choose B always B 10, 10 0,0 Assume both players discount the future at the same rate of r, 0 < r < 1. Limiting each player's strategies to the following six possibilities, S3: Choose A then mimic the other player's previous choice S4: Choose B, then mimic the other player's previous choice S5: Choose A, then choose the opposite of the other player's previous choice S6: Choose B, then choose the opposite of the other player's previous choice a. present the strategic form of this game, b. identify all pure-strategy Nash equilibria c. does repetition with these strategies "solve" the coordination dilemma that confronts the players in the single play of the above game.