Problem Set 3 Undergraduate
Economics 405
Due: October 25
- Consider a slightly modified version of Rock Paper Scissors:
| 1/2 | Rock | Paper | Scissors |
| Rock | 0, 0 | -1, 1 | 3, -1 |
| Paper | 1, -1 | 0, 0 | -1, 1 |
| Scissors | -1, 1 | 1, -1 | 0, 0 |
- What is the mixed strategy Nash equilibrium in this version of the game?
- Compare your answer with the mixed strategy equilibrium of a standard Rock Paper Scissors game (discussed in class, shown below).
| 1/2 | A | B | C |
| a | 2, 0 | -1, 0 | 2, -1 |
| b | 0, 0 | 0, 0 | 1, -2 |
| c | -1, 0 | 2, 0 | 0, -1 |
| 1/2 | Rock | Paper | Scissors |
| Rock | 0, 0 | -1, 1 | 1, -1 |
| Paper | 10, -1 | 0, 0 | -1, 1 |
| Scissors | -10, 1 | 1, -1 | 0, 0 |
How can you explain the differences in the equilibrium strategy choices?
- Lucy offers to play the following game with Charlie: “Let’s show pennies to each other, each choosing either heads or tails. If we both show heads, I pay you $3. If we both show tails, I pay you $1. If the two don’t match, you pay me $2.” Charlie reasons as follows. “The probability of both heads is ¼, in which case I get $3. The probability of both tails is ¼, in which case I get $1.
The probability of no match is ½, and in that case I pay $2, so it is a fair game.”
- Solve for the mixed strategy equilibrium for this game.
- Is Charlie’s expected payoff from this game greater than, less than, or equal to zero? What about Lucy’s expected payoff?
Draw a graph of Charlie and Lucy’s best response functions and identify the Nash equilibrium on your graph.
- Find all Nash equilibria of the following game (pure and/or mixed strategy) :
- A slave has just been thrown to the lions in the Roman Coliseum. Three lions are chained down in a line, with Lion 1 closest to the slave. Each lion’s chain is short enough that he can only reach the players immediately adjacent to him. The game proceeds as follows. First, Lion 1 decides whether or not to eat the slave. If Lion 1 has eaten the slave, then Lion 2 decides whether or not to eat Lion 1 (who is then too heavy to defend himself). If Lion 1 has not eaten the slave, then Lion 2 has no choice: he cannot try to eat Lion 1 because a fight would kill both lions. Similarly, if Lion 2 has eaten Lion 1, then Lion 3 decides whether or not to eat Lion 2.
Each lion’s preferences are fairly natural: best (4 points) is to eat and stay alive, next best (3) is to stay alive but go hungry, next (2) is to eat and be eaten, and worst (1) is to go hungry and be eaten.
- Draw the game tree with payoffs of this three-player game.
- What is the SPE of this game?
- Is there a first mover advantage to this game?
- Now modify the game so that there are four lions. Draw the game tree and find the SPE. Is the additional lion good or bad for the slave? Explain.
