# 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_1 What is the Nash equilibrium probability of Aid for the Government?
A. Between 0 and .2, inclusive.
B. Greater than .2 but less than .5.
C. Between .5 and .7, inclusive
D. Greater than .7

C. CORRECT. If the Government chooses a probability of .5 for Aid, the Pauper will not be able to take advantage of the government's choice by choosing either Work or Loaf as a pure strategy.

If the Government chooses a probability of .5 for Aid, for example, then

Pauper's Payoff (Work) = .5 (2) + .5(1) = 1.5 = Pauper's Payoff (Loaf)=.5(3) + .5 (0) =1.5.

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 to choose Aid such that the Pauper does no better from Work than from Loaf. To do that, you solve

Pauper's Payoff (Work) =X (2) + (1-X)(1) = Pauper's Payoff (Loaf)=X(3) + (1-X) (0).

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