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Developed by Prof. John Von Neumann and Oscar Morgenstern in 1928 game theory is a body of knowledge
that deals with making decisions. The approach of game theory is to seek, to determine a rival’s most profitable
counter-strategy to one’s own best moves.
Every child understands what games are. When someone overreacts, we sometimes say “it’s just a game.”
Games are often not serious. Mathematical games, which are the subject of this book, are different. It was the purpose
of game theory from its beginnings in 1928 to be applied to serious situations in economics, politics, business, and
other areas. Even war can be analyzed by mathematical game theory. Let us describe the ingredients of a
Rules - Mathematical games have strict rules. They specify what is allowed and what isn’t. Though many real-world
games allow for discovering new moves or ways to act, games that can be analyzed mathematically have a rigid set
of possible moves, usually all known in advance.
Outcomes and payoffs - Children (and grown-ups too) play games for hours for fun. Mathematical games may have
many possible outcomes, each producing payoffs for the players. The payoffs may be monetary, or they may express
satisfaction. You want to win the game.
Decision making - A game with no decisions might be boring, at least for the mind. Running a 100 meter race does
not require mathematical skills, only fast legs. However, most sport games also involve decisions, and can therefore
at least partly be analyzed by game theory.
No cheating - In real-life games cheating is possible. Cheating means not playing by the rules. If, when your chess
opponent is distracted, you take your queen and put it on a better square, you are cheating, as in poker, when you
exchange an 8 in your hand with an ace in your sleeve. Game theory doesn’t even acknowledge the existence of
cheating. We will learn how to win without cheating
Game, Play, Move: Some Definitions
The complete set of rules describes a game. A play is an instance of the game. In certain situations, called
positions, a player has do make a decision, called a move or an action. This is not the same as strategy. A
strategy is a plan that tells the player what move to choose in every possible position.
Rational Behavior is usually assumed for all players. That is, players have preferences, beliefs about the
world (including the other players), and try to optimize their individual payoffs. Moreover, players are aware that
other players are trying to optimize their payoffs.
Classification of Games
Games can be categorized according to several criteria:
How many players are there in the game? Usually there should be more than one player. However, you can
play roulette alone—the casino doesn’t count as player since it doesn’t make any decisions. It collects or
gives out money. Most books on game theory do not treat one-player games, but I will allow them provided
they contain elements of randomness.
Is play simultaneous or sequential? In a simultaneous game, each player has only one move, and all moves
are made simultaneously. In a sequential game, no two players move at the same time, and players may
have to move several times. There are games that are neither simultaneous nor sequential.
Does the game have random moves? Games may contain random events that influence its outcome. They
are called random moves.
Do players have perfect information? A sequential game has perfect information if every player, when about
to move, knows all previous moves.
Do players have complete information? This means that all players know the structure of the game—the
order in which the players move, all possible moves in each position, and the payoffs for all outcomes. Realworld games usually do not have complete information. In our games we assume complete information in
most cases, since games of incomplete information are more difficult to analyze.
Is the game zero-sum? Zero-sum games have the property that the sum of the payoffs to the players equals
zero. A player can have a positive payoff only if another has a negative payoff. Poker and chess are examples
of zero-sum games. Real-world games are rarely zero-sum.
Is communication permitted? Sometimes communication between the players is allowed before the game
starts and between the moves and sometimes it is not.
Is the game cooperative or non-cooperative? Even if players negotiate, the question is whether the results of
the negotiations can be enforced. If not, a player can always move differently from what was promised in the
negotiation. Then the communication is called “cheap talk”. A cooperative game is one where the results of
the negotiations can be put into a contract and be enforced. There must also be a way of distributing the
payoff among the members of the coalition.
Game Theory is a misnomer for Multiperson Decision Theory, analyzing the decision-making 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”:
Here, Player 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. [In
that case, we formally say that the strategies and the payoffs are common knowledge.]
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. Now he compares T and M. He realizes that, if
Player 2 plays L or m, M is better than T, but if she plays R, T is definitely better than M. Would Player 2 play R?
What would she play? To find an answer to these questions, Player 1 looks at the game from Player 2’s point of view.
He realizes that, for Player 2, there is no strategy that is outright better than any other strategy. For instance, R is the
best strategy if 1 plays B, but otherwise it is strictly worse than m. Would Player 2 think that Player 1 would play B?
Well, she knows that Player 1 is trying to maximize his expected payoff, given by the first entries as everyone knows.
She must then deduce that Player 1 will not play B. Therefore, Player 1 concludes, she will not play R (as it is worse
than m in this case). Ruling out the possibility that Player 2 plays R, Player 1 looks at his payoffs, and sees that M is
now better than T, no matter what. On the other side, Player 2 goes through similar reasoning, and concludes that 1
must play M, and therefore plays L.
This kind of reasoning does not always yield such a clear prediction. Imagine that you want to meet with a friend in
one of two places, about which you both are indifferent. Unfortunately, you cannot communicate with each other until
This situation is formalized in the following game, which is called pure coordination game:
Here, Player 1 chooses between Top and Bottom rows, while Player 2 chooses between Left and Right
columns. In each box, the first and the second numbers denote the von Neumann-Morgenstern utilities of players 1
and 2, respectively. Note that Player 1 prefers Top to Bottom if he knows that Player 2 plays Left; he prefers Bottom
if he knows that Player 2 plays Right. He is indifferent if he knows thinks that the other player is likely to play either
strategy with equal probabilities. Similarly, Player 2 prefers Left if she knows that player 1 plays Top. There is no clear
prediction about the outcome of this game.
One may look for the stable outcomes (strategy profiles) in the sense that no player has incentive to deviate
if he knows that the other players play the prescribed strategies. Here, Top-Left and Bottom-Right are such outcomes.
But Bottom-Left and Top-Right are not stable in this sense. For instance, if Bottom-Left is known to be played, each
player would like to deviate — as it is shown in the following figure:
⇐ ⇓ (0,0)
(Here, ⇑ means player 1 deviates to Top, etc.)
Unlike in this game, mostly players have different preferences on the outcomes, inducing conflict. In the
following game, which is known as the Battle of the Sexes, conflict and the need for coordination are present together.
Here, once again players would like to coordinate on Top-Left or Bottom-Right, but now Player 1 prefers to
coordinate on Top-Left, while Player 2 prefers to coordinate on Bottom-Right. The stable outcomes are again TopLeft and Bottom- Right.
Now, in the Battle of the Sexes, imagine that Player 2 knows what Player 1 does when she takes her action. This can
be formalized via the following tree:
Here, Player 1 chooses between Top and Bottom, then (knowing what Player 1 has chosen) Player 2 chooses
between Left and Right. Clearly, now Player 2 would choose Left if Player 1 plays Top, and choose Right if Player 1
plays Bottom. Knowing this, Player 1 would play Top. Therefore, one can argue that the only reasonable outcome of
this game is Top-Left. (This kind of reasoning is called backward induction.)
When Player 2 is to check what the other player does, he gets only 1, while Player 1 gets 2. (In the previous
game, two outcomes were stable, in which Player 2 would get 1 or 2.) That is, Player 2 prefers that Player 1 has
information about what Player 2 does, rather than she herself has information about what player 1 does. When it is
common knowledge that a player has some information or not, the player may prefer not to have that information —
a robust fact that we will see in various contexts.