Game Theory is a misnomer for Multiperson Decision Theory, analyzing the decisionmaking
process when there are more than one decision-makers where each agent’s payoff
possibly depends on the actions taken by the other agents. Since an agent’s preferences
on his actions depend on which actions the other parties take, his action depends on his
beliefs about what the others do. Of course, what the others do depends on their beliefs
about what each agent does. In this way, a player’s action, in principle, depends on the
actions available to each agent, each agent’s preferences on the outcomes, each player’s
beliefs about which actions are available to each player and how each player ranks the
outcomes, and further his beliefs about each player’s beliefs, ad infinitum.
Under perfect competition, there are also more than one (in fact, infinitely many)
decision makers. Yet, their decisions are assumed to be decentralized. A consumer tries
to choose the best consumption bundle that he can afford, given the prices — without
paying attention what the other consumers do. In reality, the future prices are not
known. Consumers’ decisions depend on their expectations about the future prices. And
the future prices depend on consumers’ decisions today. Once again, even in perfectly
competitive environments, a consumer’s decisions are affected by their beliefs about
what other consumers do — in an aggregate level.
When agents think through what the other players will do, taking what the other
players think about them into account, they may find a clear way to play the game.
Consider the following “game”:
2 L m R
T (1, 1) (0, 2) (2, 1)
M (2, 2) (1, 1) (0, 0)
B (1, 0) (0, 0) (−1, 1)
Here, Players 1 has strategies, T, M, B and Player 2 has strategies L, m, R. (They
pick their strategies simultaneously.) The payoffs for players 1 and 2 are indicated by
the numbers in parentheses, the first one for player 1 and the second one for player 2.
For instance, if Player 1 plays T and Player 2 plays R, then Player 1 gets a payoff of 2
and Player 2 gets 1. Let’s assume that each player knows that these are the strategies
and the payoffs, each player knows that each player knows this, each player knows that
each player knows that each player knows this,.
.. ad infinitum.
Now, player 1 looks at his payoffs, and realizes that, no matter what the other player
plays, it is better for him to play M rather than B. That is, if 2 plays L, M gives 2 and
B gives 1; if 2 plays m, M gives 1, B gives 0; and if 2 plays R, M gives 0, B gives -1.
Therefore, he realizes that he should not play B.