Game Theory Through Examples

SYMBOLIC SYSTEMS 202: The Rationality Debate (3 units) Winter Quarter 2003-2004, Stanford University Instructor: Todd Davies

Game Theory Through Examples (2/11/04)

Games against nature - decision theory for a single agent

Expected utility theory for a single agent is sometimes called the theory of "games against nature". Consider this example.

Example 1: Planning a party

Our agent is planning a party, and is worried about whether it will rain or not. The utilities and probabilities for each state and action can be represented as follows:
| | |Nature's states: | |
| | |Rain |No rain |
| | |(p=1/3) |(~p=2/3) |
|Party planner's possible actions: |Outside |1 |3 |
| |Inside |2 |2 |

The expected utility of an action A given uncertainty about a state S = Probability(S|A)*Utility(S|A) + Probability(not S|A)Utility(not S|A) Note that action A can be viewed as a compound gamble or outcome. Also, note that the probability of a state can depend on the agent's choice of action, although, in the above example, it does not.

For the party problem: EU(Outside) = (1/3)(1) + (2/3)(3) = 2.67; EU(Inside) =