Game Theory Through Examples

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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) = (1/3)(2) + (2/3)(2) = 2 Therefore, choose Outside, the action with the higher expected utility

(Noncooperative) game theory - decision theory for more than one agent, each acting autonomously (no binding agreements)

In the examples below, we'll assume two self-utility maximizing agents (or players), each of whom has complete information about the options available to themselves and the other player as well as their own and the other's payoffs (utilities) under each option.

Example 2 - Friends hoping to see each other

Consider two people, Chris and Kim.  They both enjoy each other's company, but neither can communicate with the other before deciding whether to stay at home (where they would not see each other) or go to the beach this afternoon (where they could see each other).  Each prefers going to the beach to being at home, and prefers being with the other person rather than being apart. This game can be represented by the following normal (or matrix) form:  

| | |Kim | | | | |Home |Beach | |Chris  |Home |(0,0) |(0,1) | | |Beach |(1,0) |(2,2) |

Each player has a set of strategies (={Home,Beach} for both players in this example). Specifying one strategy i for the row player (Chris) and one strategy j for the column player (Kim) yields an outcome,...
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