Economics 112
Lecture Notes
Lecture 5. Nperson Games
So far, most of the games we have studied have been greatly simplified by restricting the
story to two players. But most strategic situations involve more than two players, sometimes
many more.
This can add great complexity to game theory models. For example, Nash
equilibrium requires every player to choose a best response to the choices of all the other
players. You can imagine that as the number of players gets large it rapidly becomes
impossible to consider each possible combination of strategies and test whether any player
can benefit from switching strategies. Happily, in many interesting cases other simplifications
besides limiting the number of players will also permit straightforward analysis.
In general
we speak of
"N player" games
, where N stands for any integer, 2, 3,4,...to some potentially
very large (but finite) number.
Key terms: Nperson game; state variable; representative agent; proportional game
1. The traffic game
Twenty commuters can either take the train or drive their cars. All they care about is the
length of time it takes to get to work. The train takes 40 minutes no matter how many
passengers are riding, but the time to drive depends on how many commuters choose to
drive. Assume that payoffs are equal to the negative of the commuting time. Thus the payoff
for the train is 40. The payoff for driving is given in the following table:
Cars
Payoff
Cars
Payoff
1
22.5
11
47.5
2
25
12
50
3
27.5
13
52.5
4
30
14
55
5
32.5
15
57.5
6
35
16
60
7
37.5
17
62.5
8
40
18
65
9
42.5
19
67.5
10
45
20
70
A simple way to find an equilibrium is to test some allocations between driving and riding
the train to see if any player would wish to switch behavior. Imagine that two players drove.
Is this a Nash Equilibrium? No. Although neither driver would wish to change to riding the
train, each of the train riders would prefer to be the third car, with a payoff of 27.5 than stay
as a train rider getting 40.
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It is a Nash Equilibrium for eight players to drive. Neither drivers nor train riders can
unilaterally improve their payoff by switching to the other mode of travel.
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 Winter '08
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 Economics, Game Theory, representative

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