Consider the following prisoner's dilemma game G1:
(a) Change the payoffs so that each player's maximum payoff is 1 and their minimum payoff is zero, while making sure that they still rank lotteries over the game's outcome in the same way, assuming expected utility.
(b) What is the Nash equilibrium for this game?
Col:
q. f
Q 50, 30 5, 70
row : F 105, 20 10, 25
Prisoner's Dilemma Game G1
Take the payoffs in game G1 as representing each players' monetary
payoffs (that is, the numbers are denoted in dollars). Both players
expect the Nash equilibrium outcome. However, the row player is
now given the option of paying the column player to get the right to
make both decisions. If the Column accepts the offer, Row is allowed
to choose the outcome from {(Q, q),(Q, f),(F, q),(F, F)}. If Column
rejects the offer they will play the prisoners dilemma in G1. What is
the most amount of money that Row will offer to Column? Outline
the Nash Equilibrium that might support this amount.
Col: Row: Q 50, 30 5,70 105, 20 10, 25 Prisoner's Dilemma Game G1 (c) Take the payoffs in game G1 as representing each players' monetary payoffs (that is, the numbers are denoted in dollars). Both players expect the Nash equilibrium outcome. However, the row player is now given the option of paying the column player to get the right to make both decisions. If the Column accepts the offer, Row is allowed to choose the outcome from {(@, q), (Q, f), (F, q), (F, F)}. If Column rejects the offer they will play the prisoners dilemma in G1. What is the most amount of money that Row will offer to Column? Outline the Nash Equilibrium that might support this amount. 3. Examine the strategic games, G2, G3 and G4 (N.B. Game G4 has Three players!). In each case provide the following: (a) The strategies for each player that are strictly dominated. (b) The strategies for each player that are weakly dominated. (c) All the Pure-Strategy Nash Equilibria for the game. (d) For each Pure Strategy Nash Equilibrium, indicate whether it is strict or not. Col: Col: a b d Row: B A 53 2,1 5.5 0,6 2,2 2,2 5, 10 Row: D 7,2 3,3 4,3 E 6,4 0,4 9,9 Game G2 Game G3 Player 2: Player 2: b2 02 b2 Player 1: 41 4, 4,3 1,2,0 b1 2, 3, 2 3,3,4 Player 1: a1 2, 7, 3 7, 6, 1 b1 4, 3, 4 8, 2,3 03 b3 Player 3: Game GA 2Problemset 1 Due Thursday September 15th before 5pm 1. A decision maker who obeys the von-Neumann-Morgenstern axioms is choosing over 5 potential prizes X = {x1, 12, 23, DA, X5} . We know for sure that she ranks them according to We normalize the best and worst prizes to have payoffs v() = 100 and "(25) =0. Other than this all we know is the following: . The lottery A which gives her 23 with certainty is ranked as in- different to the lottery B which gives a with 45 percent proba- bility and 25 with 55 percent probability. [We can write this as (0, 0, 1, 0, 0) = PA ~ PB = (0.45, 0, 0, 0, 0.55).] . The lottery that gives 2 with certainty is ranked as indifferent to the lottery which gives $1 with 60 percent probability, $2 with 20 per- cent probability and as with 20 percent probability. [(0, 1, 0, 0,0) ~ (0.6, 0.2, 0, 0, 0.2)] . The lottery that gives z2 with probability one-third, and 24 with probability two-third is strictly preferred to the lottery which gives To with probability two-thirds and T4 with probability one third. [(0, }, 0, 3, 0) > (0, 0, 3, 5,0).] (a) What values do v(13) and v(12) take? (b) What do we know about the value of v(14)? (c) Consider two lotteries C and D, where po = ( 10 : 10 : 3: 3: 2) PD = (10: 10: 10: 10: 10). What do we know about how the decision maker will rank C and D? 2. Consider the following prisoner's dilemma game G1: (a) Change the payoffs so that each player's maximum payoff is 1 and their minimum payoff is zero, while making sure that they still rank lotteries over the game's outcome in the same way, assuming expected utility. (b) What is the Nash equilibrium for this game
