Two players are bargaining over a three period bargaining model as discussed in class with player 1 making offers in rounds 1 and 3. Player 2 makes an offer in round 2 only. Each player has a common discount factor delta. The two players are bargaining to split $20. They have three time periods available to them for their bargaining game. At the end of round 3, if no agreement has been reached then player 1 receives $2 and player 2 receives $1 and the rest of the money
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- two players, a and b are playing an asymmetrical game. there are n points on the game board. each turn player a targets a pair of points and player b says whether those two points are connected or unconnected. a can target each pair only once and the game ends when all pairs have been targeted. player b wins if a point is connected with all other points on the very last turn, while player a wins if any point is connected with all other points on any turn but the very last one or if no point is connected to all other points after the last turn. for what values of n does either player have a winning strategy?Two players bargain over $20. Player 1 first proposes a split of(n, 20 - n), where n is an integer in {0, 1, ..., 20}. Player 2 can either accept or reject this proposal. If player accepts it, player 1 obtains $n and player 2 obtains $(20 - n). If player 2 rejects it, the money is taken away from them and both players will get $0. Find one Nash equilibrium that is NOT a subgame perfect Nash equilibrium. Explain why it is a Nash equilibrium and why it is not subgame perfect.Amir and Beatrice play the following game. Amir offers an amount of money z € [0, 1] to Beatrice. Beatrice can either accept or reject. If Beatrice accepts, then Amir receives 1 - z dollars and Beatrice receives a dollars. If Beatrice rejects, then both receive no money. As in the Ultimatum Game, Amir cares only about maximizing the amount of money he receives. Beatrice, on the other hand, detests Amir, and therefore cares about the amount of money that both receive: if Amir receives y dollars and Beatrice receives a dollars, then Beatrice's payoff is a-ay where a > 0. (a) Find all pure strategy Nash equilibria of the game in which the two players choose simultaneously (thus Beatrice accepts or rejects without seeing Amir's offer). Solution: The NE are the strategy profiles in which Beatrice rejects and Amir's offer a satisfies 2-a(1-x) ≤0, i.e. r ≤a/(1+a). (b) Find all subgame perfect equilibria of the sequential game in which Amir first makes the offer and Beatrice observes the offer…
- 2. Consider the following "centipede game." The game starts with player 1 choosing be- tween terminate (T) and continue (C). If player 1 chooses C, the game proceeds with player 2 choosing between terminate (t) and continue (c). The two players choose be- tween terminate and continue in turn if the other player chooses continue until the terminal nodes with (player l's payoff, player 2's payoff) are reached as shown below. TTTT Player 1 Player 2 Player 1 Player 2 (3, 3) t (1, 1) (0, 3) (2, 2) (1, 4) (a) List all possible strategies of each player. (b) Transform the game tree into a normal-form matrix representation. (c) Find all pure-strategy Nash equilibria. (d) Find the unique pure-strategy subgame-perfect equilibrium.Consider the following simultaneous game: Player 1 U D Player 2 L 30,20 -10, -10 R -10, -10 20,30 Suppose player 1 plays a mixed strategy in which she plays U 25% of the time and D 75% of the time, and player 2 plays a mixed strategy in which she plays L 25% of the time and R 75% of the time. This pair of strategies ✓a Nash equilibrium. Player 1's expected payoff from playing U (when player 2 plays the mixed strategy above) isIn a gambling game, Player A and Player B both have a $1 and a $5 bill. Each player selects one of the bills without the other player knowing the bill selected. Simultaneously they both reveal the bills selected. If the bills do not match, Player A wins Player B's bill. If the bills match, Player B wins Player A's bill. a. Develop the game theory table for this game. The values should be expressed as the gains (or losses) for Player A. b. Is there a pure strategy? Why or why not? Determine the optimal strategies and the value of this game. Does the game favor one player over the other? d. Suppose Player B decides to deviate from the optimal strategy and begins playing each bill 50% of the time. What should Player A do to improve Player A’s winnings? Comment on why it is important to follow an optimal game theory strategy. с.
- 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.10 1. Consider the game where initially She chooses between "Stay Home" and "Go Out". If She chooses "Stay Home" then She gets 2 and He gets 0. If She chooses "Go Out" then they each simultaneously choose "Movie" or "Concert" where the payoffs are 0,1 or 3 as in the Battle of the Sexes Game. What are the subgame perfect Nash Equilibria of this game ?PLAYER B LEFT RIGHT UP 5 FOR A, 30 FOR B 10 FOR A, 12 FOR B PLAYER A DOWN -2 FOR A, 10 FOR B 8 FOR A, 15 FOR B In the above game, the players are seeking to maximize the number they recieve. They choose at the same time. What is the Nash equillibrium? Player A will choose UP and player B will choose LEFT Player A will UP and player B will choose RIGHT Player A will choose DOWN and player B will choose LEFT Player A will choose DOWN and player B will choose RIGHT Player A will choose LEFT and player B will choose UP Player A will choose LEFT and player B will choose DOWN Player A will choose RIGHT and player B will choose UP Player A will choose RIGHT and player B will choose DOWN
- 14. You have baked a cake, but your two dear daughters won't stop fighting on who gets the biggest slice. To settle the dispute, to ask your dear daughter one (DD1) to cut the cake and your dear daughter two (DD2) to choose which piece she wants. (a) Draw the extensive form of the game. Let dear daughter one's strategies be "Cut Evenly" or "Cut Unevenly"; depending on what is on the platter, dear daughter two's strategies might in- clude "Take Big Slice", "Take Small Slice", or "Take Equal Slice". Assign payoffs to dear daughter one and dear daughter two that grow with the size of the slice that they receive. (b) Use backward induction to find the equilibrium outcome of this game. (c) Is the promise to take a small slice by DD2, if DD1 cuts unevenly, credible? Explain carefully. (d) After the rules are announced, dear daughter two says "It is not fair! I want to be the one who gets to cut the cake, not the one who chooses the slice!". Is dear daughter two's complaint valid? You are…Q5 A and B are sharing a room with a TV and a DVD player. DVD's can be rented for the weekend for a cost of $5. They would each get $4 worth of enjoyment from a weekend movie, so if both of them rent a movie a on a particular weekend they can each get $8 worth of enjoyment at a cost of $5 rental. (a) Model this as a strategic form game with both players having the strategies Rent (R) and No Rent (N), (b) What is the equilibrium if they share the room for one weekend only? (e) Draw the extensive form of the game if they stay together for two weeks, and identify the subgame perfect equilibria. (d) Explain whether or not the outcome of the game will change if they stay together for 10 weeks.Anashequilibriumisastrategyprofilesuchthat: a)Both players are playing a best response strategy b)Both players must be playing a dominant strategyc)Both players achieve their highest possible payoff d)a) and c) e)b) and c)