7. Solving for dominant strategies and the Nash equilibrium Suppose Antonio and Trinity are playing a game that requires both to simultaneously choose an action: Up or Down. The payoff matrix that follows shows the earnings of each person as a function of both of their choices. For example, the upper-right cell shows that if Antonio chooses Up and Trinity chooses Down, Antonio will receive a payoff of 4 and Trinity will receive a payoff of 4. Antonio Up Down Up 8,3 5,3 Trinity Down 4,4 5,4 Down Up
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- Suppose Edison and Hilary are playing a game in which both must simultaneously choose the action Left or Right. The payoff matrix that follows shows the payoff each person will earn as a function of both of their choices. For example, the lower-right cell shows that Edison chooses Right and Hilary chooses Right, Edison will receive a payoff of 3 and Hilary will receive a payoff of 7. Hilary Left Right Left 4, 6 6, 8 Edison Right 7, 5 3, 7 The only dominant strategy in this game is for to choose The outcome reflecting the unique Nash equilibrium in this game is as follows: Edison chooses and Hilary chooses7. Solving for dominant strategies and the Nash equilibrium Suppose Dmitri and Frances are playing a game in which both must simultaneously choose the action Left or Right. The payoff matrix that follows shows the payoff each person will earn as a function of both of their choices. For example, the lower-right cell shows that if Dmitri chooses Right and Frances chooses Right, Dmitri will receive a payoff of 7 and Frances will receive a payoff of 6. Frances Left Right Left 4, 3 6, 4 Dmitri Right 6, 7 7, 6 to choose The only dominant strategy in this game is for and Frances chooses The outcome reflecting the unique Nash equilibrium in this game is as follows: Dmitri chooses vConsider the following two-player game.First, player 1 selects a number x≥0. Player 2 observes x. Then, simultaneously andindependently, player 1 selects a number y1 and player 2 selects a number y2, at which pointthe game ends.Player 1’s payoff is: u1(x; y1) = −3y21 + 6y1y2 −13x2 + 8xPlayer 2’s payoff is: u2(y2) = 6y1y2 −6y22 + 12xy2Draw the game tree of this game and identify its Subgame Perfect Nash Equilibrium.
- Consider the payoff matrix for a game depicted below. Player 1 selects the row and Player 2 selects the column. Up Down Left 1, -1 -1, 1 Right -1, 1 1, -1 What is (are) the Nash equilibrium (equilibria)? Question 18Answer a. Player 1 plays right; Player 2 plays down b. Player 1 plays left; Player 2 plays down c. Player 1 plays down; Player 2 plays left d. Player 1 plays right; Player 2 plays up e. Player 1 plays up; Player 2 plays left f. There is no Nash equilibrium g. Player 1 plays down; Player 2 plays right h. Player 1 plays up; Player 2 plays right i. Player 1 plays left; Player 2 plays upSuppose Antonio and Trinity are playing a game that requires both to simultaneously choose an action: Up or Down. The payoff matrix that follows shows the earnings of each person as a function of both of their choices. For example, the upper-right cell shows that if Antonio chooses Up and Trinity chooses Down, Antonio will receive a payoff of 7 and Trinity will receive a payoff of 5. Trinity Up Down Up 4,8 7,5 Antonio Down 3,2 5,6 In this game, the only dominant strategy is for to choose The outcome reflecting the unique Nash equilibrium in this game is as follows: Antonio chooses, and Trinity chooses Grade It Now Save & Continue Continue without saving @ 2 F2 #3 80 Q F3 MacBook Air 44 F7 Dll F8 44 F10 74 $ 4 05 Λ & % 5 6 7 8 * 0 Q W E R T Y U 1 A N S X 9 0 -O O D F G H J K L on را H command C > B N M Λ - - P [ H Λ command optiConsider a game in which two players simultaneously pick an integer from 1 through 10. Let s1 denote choice of player 1, and let S2 denote the choice of player 2. If s1 7 82 then player 1's payoff is: Si + 82 2 and player 2's payoff is 10 – ,*. Si + 82 2 If s1 = 82 then each player gets 5. How many pure strategy Nash equilibria does the game have (you will get full points if you are within 2 of the correct answer)? Numerical answer
- Suppose Yakov and Ana are playing a game in which both must simultaneously choose the action Left or Right. The payoff matrix that follows shows. the payoff each person will earn as a function of both of their choices. For example, the lower-right cell shows that if Yakov chooses Right and Ana chooses Right, Yakov will receive a payoff of 6 and Ana will receive a payoff of 6. Yakov Left Left 2,3 Right 3,7 Ana Right 4,4 6,6 The only dominant strategy in this game is for to choose The outcome reflecting the unique Nash equilibrium in this game is as follows: Yakov chooses and Ana choosesConsider the following game. There are two payers, Player 1 and Player 2. Player 1 chooses a row (10, 20, or 30), and Player 2 chooses a column (10/20/30). Payoffs are in the cells of the table, with those on the left going to Player 1 and those on the right going to player 2. Suppose that Player 1 chooses his strategy (10, 20 or 30), first, and subsequently, and after observing Player 1’s choice, Player 2 chooses his own strategy (of 10, 20 or 30). Which of the following statements is true regarding this modified game? I. It is a simultaneous move game, because the timing of moves is irrelevant in classifying games.II. It is a sequential move game, because Player 2 observes Player 1’s choice before he chooses his own strategy.III. This modification gives Player 1 a ‘first mover advantage’. A) I and IIB) II and IIIC) I and IIID) I onlyE) II onlySuppose Carlos and Deborah are playing a game in which both must simultaneously choose the action Left or Right. The payoff matrix that follows shows the payoff each person will earn as a function of both of their choices. For example, the lower-right cell shows that if Carlos chooses Right and Deborah chooses Right, Carlos will receive a payoff of 6 and Deborah will receive a payoff of 5. Deborah Left Right Carlos Left 8, 4 4, 5 Right 5, 4 6, 5 The only dominant strategy in this game is for to choose . The outcome reflecting the unique Nash equilibrium in this game is as follows: Carlos chooses and Deborah chooses .
- 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) is3. Solving for dominant strategies and the Nash equilibrium Suppose Charles and Dina are playing a game in which both must simultaneously choose the action Left or Right. The payoff matrix that follows shows the payoff each person will earn as a function of both of their choices. For example, the lower-right cell shows that if Charles chooses Right and Dina chooses Right, Charles will receive a payoff of 3 and Dina will receive a payoff of 8. Dina Left Right Left 3,7 2,6 Charles Right 4,5 3,8 The only dominant strategy in this game is for to choose The outcome reflecting the unique Nash equilibrium in this game is as follows: Charles chooses and Dina choosesConsider the following sequential game. Wally first chooses L or H. Having observed Wally's choice, Elizabeth chooses between A and F. The payoffs are as follows. If Wally chose L and Elizabeth chose A, the payoffs are 30 to Wally and 20 to Elizabeth. If Wally chose L and Elizabeth F, the payoffs are 40 to Wally and 10 and to Elizabeth. If Wally decides to opt for H and Elizabeth A, the payoffs are 10 and 2 to Wally and Elizabeth, respectively. Finally, if Wally opts for H and Elizabeth F, the payoffs are 35 toWally and 5 to Elizabeth. What is the outcome in the subgame perfect equilibrium of this game? (L,F) (H,A) (H,F) and (L,F) (L,A) and (H,F) (H,F)