# 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

A. CORRECT. If the Pauper chooses to work with probability .2, then the Government will be just indifferent between Aid and No Aid. This will mean that the government is willing to randomize, rather than choosing a pure strategy, and that will permit a Nash equilibrium in mixed strategies.

Government's Payoff (Aid) =.2 (3) + (1-.2)(-1) = -.2 =Government's Payoff (No Aid)=X(-1) + (1-X) (0) = -.2.

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).

This reduces to 3X -1 +X = -X, so 5X = 1, so X =.2.