Player 1 Cooperate (C) Defect (D) If the game has a dominant strategy, what is it? There is none. If the game has a Nash equilibrium in pure strategies, what is it? There is none. Cooperate (C) 3,3 8,0 O Cooperate (C) is a dominant strategy for both players. Defect (D) is a dominant strategy for both players. Cooperate (C) is a dominant strategy for 1, and Defect (D) is a dominant strategy for 2. OC, C is the only Nash equilibrium. OD, D is the only Nash equilibrium. OC, C and D, D are both Nash equilibria. Player 2 Defect (D) 0,8 1, 1
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- Player 1 Cooperate (C) Defect (D) Cooperate (C) 3,3 8,0 Player 2 Defect (D) 0,8 1,1 In general, a combination of strategies is a Nash equilibrium if ... Every player is choosing a best response against the other players' strategies. Every player has a positive payoff. The players maximize the sum of their payoffs. The players choose identical strategies. If the game is repeated, which cooperative actions could benefit both players? O Both players choose C. Player 1 chooses C, Player 2 chooses D. O Player 1 chooses D, Player 2 chooses C. Both players choose D.Which is the correct mixed strategy Nash equilibrium for the below game? A B Confess Confess (3,2) Doesn't (0,0) confess a. (2/5); (3/5) b. (1/4); (3/4) c. (1/2); (1/2) d. (1/3); (2/3) Doesn't confess (0,0) (2,3)B block allow 5,9 3, 11 high A 6,2 4,4 low In this normal form game, there is a Nash equilibrium where A chooses high; block high; allow low; block low; allow and B chooses
- Two business partners jointly own a firm and share equally the revenues. They individually and simultaneously decide how much effort to put into the firm. Let s₁ and s2 denote the effort choices of partner 1 and partner 2, respectively. Assume si € [0, 4]. The cost of effort is given by s? for i E {1, 2}. The firm's revenue is given by 4(81 +82 + bs182) where 0 ≤ b ≤ 1. (Note that the parameter 6 reflects the synergies between the effort levels. b> 0 implies that the more one partner works, the more productive the other partner is.) The payoffs for partners 1 and 2 are: u₁ (81, 82) u2 (81, 82) - = 1 [4(81 +82 +68182)] − 8² - 1 [4(81 +82 + bs182)] – $²(Demand Game) Two players can divide ten dollars. Find the Nash equilibria of the following two games. (a) The players simultaneously submit demands, numbers in {0, 1, ... ,10}. If the sum of the demands is at most 10, each player gets what she demands. Otherwise both get 0. (b) As in the previous part, except that if the sum of the demands exceeds 10, then (i) if the demands differ then the player who demands less gets her demand and the other player gets the rest, and (ii) if the demands are the same then each player gets 5. (Hint- write the best response functions).Consider the following game. Firm 1 can implement one of two actions, A or B. Firm 2 observes the action chosen by Firm 1 and then decides whether to fight it or not. (-10, 20) F2 Fight A Don't fight -(30, 10) Firm 1 (-10, 0) Fight B Don't fight -(20, 15) (a) Consider the following strategy profile: Firm 1 chooses A; Firm 2 chooses fight if A, and fight if B. • this strategy profile is [Select] (b) Consider the following strategy profile: Firm 1 chooses B; Firm 2 chooses fight if A, and don't fight if B. • this strategy profile is [Select] O (c) Consider the following strategy profile: Firm 1 chooses B; Firm 2 chooses don't fight if A and don't fight if B. • this strategy profile is [Select] 00 F2 ()
- (b) ROWENA up Answer 1: up Answer 2: In this simultaneous move game, the iterated dominance equilibrium is (Rowena's . Colin's left left Answer 3: ). This game has two Nash equilibria one of which is the iterated dominance equilibrium. The other Nash equilibrium, which is not the IDE, is (Rowena's down Colin's right down Up Down Answer 4: Left right 0,0 0,0 COLIN Right 0,0 1,1GAME 5 Player B B1 B2 Player A A1 7,3 | 5, 10 A2 3, 8| 9, 6 In Game 5 above, O Neither player has a dominant strategy. O Player B has a dominant strategy. O Player A has a dominant strategy. O Both players have dominant strategies.8) Find the mixed strategy Nash equilibrium of the following normal form game. Player 2 T1 T2 T3 2, 3 3, 5 1, 1 Player 1 S2 1, 4 4, 3 0, 5 Player 1 attaches probability (S1, S2) = () and Player 2 attaches probability (T1, T2, T3) = ( ) Player 1 attaches probability (S1, S2) = (.) and Player 2 attaches probability (T1, T2, T3) = (qi, 42, 1 – q1 – 92) where q1 , and 0 < q2 S %3D Player 1 attaches probability (S1, S2) = (G,;) and Player 2 attaches probability (T1, T2, T1) = (qı.42, 1 – q1 – 42) where 0 < qi <, and q2 = 3. Player 1 attaches probability (S1, S) = (;, -) and player 2 attaches probability (T1, T2, T3) = (1.42, 1- q1- 42) where 0 s qı s and q2 =
- (a) Consider a ROCK PAPER SCISSOR game. Two players indicate either Rock, Paper or Scissor simultaneously. The winner is determined by: Rock crushes Scissors, Paper covers Rock, and Scissor cut Paper. In the case of a tie, there is no payoff. In the case of a win, the winner collects 5 dollars. Write the payoff matrix for this game. (b) Find the optimal row and column strategies and the value of the matrix game. 3 2 4 -2 1 -4 5し(5,3) b I(2,2) も(0,0) (4,12) a (12.4) (0,0) i). List all subgame pertect Nash equilibria and name one Nash eqvilibrivm that is not subgame pertect i). How many strategies does playot and player 2 have?GAME ZZZ B1 Player B A1 30, 30 Player A A2 20, 40 B2 40, 20 35, 35 In the Game ZZZ (see table above), all payoffs are listed with the row player's payoffs first and the column player's payoffs second. In this game, neither player has a dominant strategy. the Nash equilibrium does not maximize the total payoff. there is no Nash equilibrium. the Nash equilibrium maximizes the total payoff.