# An Answer for the Self-test on Mixed Strategies

GAME 3: THE WELFARE GAME
```                                         PAUPER
Work         Loaf
Aid           3,2           -1,3
GOVERNMENT
No  Aid       -1,1           0,0

```
``The Welfare Game'' models a government that wishes to aid a pauper if he searches for work but not otherwise, and a pauper who searches for work only if he cannot depend on government aid. This is a well-known problem in public policy, called ``The Samaritan's Dilemma'' by Tullock (1983, p. 59), who attributes it to James Buchanan. I use it in Chapter 3 of Games and Information . The same problem arises on a private level when parents decide how much to help a lazy child.

3_2 What is the Nash equilibrium probability of Work for the Pauper?
A. Between 0 and .2, inclusive.
B. Greater than .2 but less than .5.
C. Between .5 and .7, inclusive
D. Greater than .7

B. Try again. If the Pauper chooses to Work with such a high probability, then the Government will choose Aid as a pure strategy. That, in turn, would make the Pauper want to deviate to Loaf, so it couldn't be part of a Nash equilibrium.

If the Pauper chooses to Work with probability .4, for example, Government's Payoff (Aid) =.4 (3) + (1-.4)(-1) = .6 > Government's Payoff (No Aid)=.4(-1) + (1-.4) (0) = -.4.

To find a Nash equilibrium, in which each player would not deviate from his equilibrium mixing probability, you need to choose a mixing probability X for the Government such that the Pauper does no better from Work than from Loaf. To do that, you solve

Government's Payoff (Aid) =X (3) + (1-X)(-1) = Government's Payoff (No Aid)=X(-1) + (1-X) (0).