Player 1 and Player 2 have formed a partnership (e.g., a law firm). Suppose both players simultaneously invest some effort and split the joint return equally. Let x and y denote the amounts of effort that Player 1 and Player 2, respectively, invest. This will result in a disutility of x²/2 for Player 1 and a disutility of y/2 for player 2. However, it will result in a joint return of (x+y) which the players split evenly. a) Left to their own devices, what effort levels will the players' choose? Show calculations.
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- Players 1, 2, and 3 are playing a game in which the strategy of player i isdenoted yi and can be any nonnegative real number. The payoff function for player 1 is V1(y1,y2,y3) = y1 + y1y2 - (y1)2,for player 2 is V2(y1,y2,y3) = y2 + y1y2 - (y2)2,and for player 3 is V3(y1,y2,y3) = (10 - y1 - y2 - y3)y3.These payoff functions are hill shaped. Find a Nash equilibrium. (Hint: Thepayoff functions are symmetric for players 1 and 2.)(ii) A mixed strategy profile (p, q) is one in which p = (p,P2.... P) is the mixed strategy of player 1, and q- (g1, q2,..q4) is the mixed strategy of player 2. Show that if p, >0 in a Nash equilibrium profile (p*, q*), the player 2 must also play i with strictly positive probability q'; > 0. (State clearly any theorem you use to show this. You are not required to justify the theorem.) %3DTwo players play the following game for infinite times. For the player to continue to cooperate what would be the ranges of their discount factor, δ_1 and δ_2, respectively? cooperate betray cooperate (10,20) (-25,30) betray (15, -22) (-12, -18)
- (b) Consider the simultaneous-move game below with two players, 1 and 2. Each player has two pure strategies. If a player plays both strategies with strictly positive probability, we call it a strictly mixed strategy for that player. Show that there is no Nash equilibrium in which both 1 and 2 play a strictly mixed strategy. Player 2 b₁ b₂ Player 1 a₁ 3,0 0,1 a2 2,1 2,1A recently discovered painting by Picasso is on auction at Sotheby's. There are two main bidders Amy and Ben {1,2}. Bidding starts at £10M but the value of the painting is certainly not more than £20M. Each bidder's valuation v; is independently and uni- formly distributed on the interval [10M, 20M], and this is common knowledge among the players: A bidder knows their own valuation but not of their opponent. Consider an auction where an object is allocated to the highest bidder but the price paid by the bidder is determined randomly. With probability 3/4, the bidder pays their own bid, and with probability 1/4 the bidder pays the losing bid. The person bidding lowest pays nothing. If the bids are equal, each bidder gets the object with probability one-half, and in this case, pays their bid. Suppose that bidder 1 assumes that bidder 2 will bid a constant fraction, 7, of bidder 2's valuation (and similarly, bidder 2 assumes bidder 1 will bid the same constant propor- tional value of…R (3,2) (0,0) (2,4) (5,4) (0,0) In the extensive form representation of the game between Player 1 and Player 2, Player 1 moves first and chooses L or R. If Player 1 chooses R the game ends, if Player 1 chooses L then Player 1 and 2 play a simultaneous move game. The game has strategy Subgame Perfect Nash Equilibria (SPNE). The maximum payoff Player 2 gets in a SPNE is pure strategy Nash equilibria and pure (Please, enter only numerical answers like: 1, 2, 3, ...)
- Is it possible that Step. 3 contains a contradiction? As "If Player 1 plays Ball (B): Player 2 meets (M) if the expected payoff of meeting is greater than the expected payoff of avoiding." and then, "If Player 1 has Dinner (D): Player 2 meets (M) if the expected payoff of meeting is greater than the expected payoff of avoiding." The same arises in the one for player 2. Is for both the second sentence meant to be "smaller than"? If not, please elaborate.In Homework 1, we saw that there is no strict dominant strategy equilibrium in an inspection game. Thus, the players must randomize and play a mixed strategy in equilibrium. Suppose the agent shirks with probability p₁ and works with probability (1 - p₁), while the principal inspects with probability p2 and does not inspect with probability (1-P2). Find the mixed strategy equilibrium for this game. I NI S (0, -k) (w, -w) W (w c, p-w-k)| (wc, p-w) Table 1: Payoff matrix for the inspection gameSuppose there are two players playing a game with east or west and south and nerth ways. Find the expected Nash equilibrium by using the concept of probabilities. Player X Left[L) Right|R) Player Y Up(U) (5,6) (0,8) (4,6) Down[D) (0,9)
- Consider the following normal form game Player 2 L C Player 1 U 5,4 4,5 A 3,8 UMD 8,3 8,0 8.8 R 5,5 6,3 0,8There is a project for which player 1 can exert effort e ≥ 0 that costs her c(e) = 0.5e2. If player 1 and player 2 can come to an agreement, then a total value of v(e) = e is produced, which can be allocated between the two players. Effort also produces a value y(e) = ke, where k = [0, 1], that player 1 can obtain for herself if player 1 and 2 fail to agree. The game has three stages: (I stage) Player 1 chooses effort e ≥ 0; (II Stage) Player 2 observes e and chooses and effort level a € [0, 1]; (III stage) Player 1 observes a and either agrees (a) or rejects (r) the offer. If player 1 accepts the offer, then her payoff is ae - 0.5e² and player 2's payoff (1 − a)e. If she rejects the offer, then player 1's payoff is ke - 0.5e² and player 2's payoff is zero. • What is the subgame perfect equilibrium effort choice? •Suppose that player 2 can choose k = [0, 1] before the above game is played. What would he choose? What would be the equilibrium effort? Suppose now that the government,…David wants to auction a painting, and there are two potential buyers. The value for eachbuyer is either 0 or 10, each value equally likely. Suppose he offers to sell the object for $6, and the two buyers simultaneously accept or reject. If exactly one buyer accepts, the object sold to that person for $6. If both accept, the object is allocated randomly to the buyers, also for $6. If neither accepts, the object is allocated randomly to the bidders for $0. (a) Identify the type space and strategy space for each buyer. (b) Show that there is an equilibrium in which buyers with value 10 always accept. (c) Show that there is an equilibrium in which buyers with value 10 always reject.