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Microeconomics 1 Block 4 Game Theory Lecture Notes 20/21
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Microeconomics 1 Block 4 Game Theory Lecture Notes 20/21
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Micro Block 4: Game Theory4.1 Strategic games
Game theory is the study of interacting decision makers. Decision makers make decisions that
take into account the other agent’s actions and responses; thus the decision makers are making
strategic decisions. The environment is therefore more complex than what we saw before. Game
theory will help us study the strategic behaviour of economic agents in these more complex
environments.
Strategic games
A strategic games refers to any situation in which decision makers make simultaneous strategic
decisions, i.e. their decisions take into account the actions and responses of the other decision
makers. Therefore, a strategic game consists of a set of players (participants) a set of actions (for
each player there is a set of possible actions/moves) and payo s (payo s specify for each player
the value of each possible outcome of the game).
Payo matrix
Whenever there are only two players and a limited number of actions for each player, it is possible
to represent a strategic game using a payo matrix as in gure 54. The possible actions of the rst
player are represented in the rows, and the possible actions of the second player are represented
in the columns. The payo s are shown in the corresponding cells.
In the game in gure 54, the set of players is A and B. The set of actions for A is U = up and D =
down and the set of actions for B is L = left and R = right. There are four possible outcomes in this
game, and the payo s are shown in the corresponding cells (the rst number is the payo to
player A and the second number is the payo to player B).
Figure 54: the payo matrix
Strategy
When playing the game, players can devise strategies. A strategy is a rule or plan of action for
playing the game.
Best response function
Suppose there are two players, A and B. A can choose any action r (we denote the choice r
because A’s choices are represented in the rows of the payo matrix) in the set r1, …, rR. Similarly,
B can choose any action c in the set c1, …, cC.
The best response function br(c) de nes the best response for player A to any choice the other
player makes. Similarly, the best response function bc(r) de nes the best response for player B to
any choice the other player makes.
Nash equilibrium
Game theory seeks to determine the likely outcome of a game. To identify likely outcomes, we use
the concept of Nash equilibrium. An outcome of the game is a Nash equilibrium if the strategy
played by each player is a best reply to the strategies played by the other players. The compelling
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, property of Nash equilibria is that they are self-enforcing: given the strategies chosen by the other
players, each player will stick to his chosen strategy.
Using the best response functions, in a two-player strategic game, we can de ne a Nash
equilibrium as a pair of strategies (r*, c*) such that c* = bc(r*) and r* = br(c*).
EXAMPLE Consider the game in gure 55. There are two Nash equilibria: (U, L) and (D, R).
Figure 55: a strategic game example
Dominant strategy
A dominant strategy for a given player is a strategy that is best no matter what the other players
do. In the example in gure 45, U is a dominant strategy for A, and L is a dominant strategy for B.
Dominant strategies make it easier to determine the likely outcome of a game, since we know that
player who have a dominant strategy will play it.
When both players (in a 2-players game) have a dominant strategy, those strategies de ne the
Nash equilibrium of the game. If only one player (in a 2-players game) has a dominant strategy, the
Nash equilibrium is de ned by the other player’s best response to that strategy.
The prisoners’ dilemma
The prisoners’ dilemma is an important game to illustrate the concepts we have introduced so far,
and to discuss the relationship between the Nash equilibrium and Pareto e ciency.
A and B are arrested by the police for illegally producing meth. They are interrogated in two
separate rooms. They have two possible actions: confess their crime (and throw their partner
under the bus) or deny. If both deny, the both get a small jail sentence (1 year). If one player
confesses and the other denies, the play who confesses gets no jail time (0 years), and the other
one will go to prison for a long time (6 years). If both confess, they both get fairly long jail time (3
years), which is less than 6 because they have both confessed.
Whatever player B plays, player A’s best response is to play confess: confess is a dominant
strategy for player A. The same is true for player B. The only Nash equilibrium in this game is
therefore (C, C). If the outcome of the game is (C, C) (which is what we can expect, then we can
see that both players could actually be better o if they player (D, D). Thus it is clear from this
example that a Nash equilibrium is not necessarily Pareto e cient.
Figure 56: the prisoners’ dilemma
Pure and mixed strategy
So far, we have implicitly assumed that players can only devise pure strategies, i.e. they play
purely one of the possible actions. Players can also devise mixed strategies, in which players
choose a probability distribution over the possible actions.
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