Game Theory in Economics : Game Theory in Economics
TOPICS TO BE DISCUSSED : 1 TOPICS TO BE DISCUSSED Introduction to game theory
History of Game Theory
Payoff Matrix
Basic Strategies
Applications
Limitations and Problems
Game Theory - Introduction : 2 Game Theory - Introduction Game theory is a branch of applied mathematics, which pertains to study of strategic interaction between individuals in situations called GAMES
In other words, game theory attempts to mathematically capture behavior in strategic situations, in which an individual's success in making choices depends on the choices of others
Applications of game theory attempt to find equilibria in strategic situations / games - sets of strategies in which individuals are unlikely to change their behavior
Although some developments occurred before it, the field of game theory came into being with the 1944 book “Theory of Games and Economic Behavior” by John von Neumann
This theory was developed extensively in the 1950s by many scholars…A Coalition-Proof Nash Equilibrium (CPNE) (similar to a Strong Nash Equilibrium) occurs when players cannot do better even if they are allowed to communicate and collaborate before the game.
History of Game Theory : 3 History of Game Theory 1913 - E. Zermelo provided the first theorem of game theory
asserts that chess is strictly determined
1928 - John von Neumann proved the minimax theorem
1944 - John von Neumann / Oskar Morgenstern’s wrote
"Theory of Games and Economic Behavior”
1950-1953, John Nash describes Nash equilibrium
1972 - John Maynard Smith wrote
“Game Theory and The Evolution of Fighting”
Game Theory – Basics : 4 Game Theory – Basics Ingredients
a) Number of players
b) Feasible strategy set (actions) for each player
c) Payoff with each strategy combination
A game consists of a set of players, a set of moves (or strategies) available to those players, and a specification of payoffs for each combination of strategies
Two basic representations
a) Normal Form
b) Extensive Form (nodes and branches)
Extensive form : 5 Extensive form The extensive form can be used to formalize games with
some important order. Games here are often presented
as trees (as pictured to the left). Here each vertex (or node)
represents a point of choice for a player.
In the game pictured here, there are two players. Player 1
moves first and chooses either F or U. Player 2 sees Player 1's
move and then chooses A or R. Suppose that Player 1 chooses
U and then Player 2 chooses A, then Player 1 gets 8 and
Player 2 gets 2. The extensive form can also capture
simultaneous-move games and games with incomplete
information. To represent it, either a dotted line connects different
vertices to represent them as being part of the same information set
(i.e., the players do not know at which point they are), or a closed
line is drawn around them. Main article: Extensive form game
An extensive form game
Representation in Normal Form : 6 Representation in Normal Form The normal (or strategic form) game is usually represented by a matrix which shows the players, strategies, and payoffs (refer to Fig.1)
In this example, there are two players; one chooses the row and the other chooses the column
Each player has two strategies, which are specified by the number of rows and the number of columns
As for the payoffs which are specified in the matrix, the first number is the payoff received by the row player (Player 1 in our example); the second is the payoff for the column player (Player 2 in our example)
Suppose that Player 1 plays Up and that Player 2 plays Left; then Player 1 gets a payoff of 4, and Player 2 gets 3 Fig. 1
Slide 8: 7 In game theory, the Nash equilibrium (named after John Forbes Nash, who proposed it)
is a solution concept of a game involving two or more players, in which each player is
assumed to know the equilibrium strategies of the other players, and no player has
anything to gain by changing only his or her own strategy (i.e., by changing unilaterally).
If each player has chosen a strategy and no player can benefit by changing his or her
strategy while the other players keep theirs unchanged, then the current set of strategy
choices and the corresponding payoffs constitute a Nash equilibrium.
Nash Equilibrium – Overview : 8 Nash Equilibrium – Overview Many equilibrium concepts have been developed pertaining to game theory, the most famous of which being the Nash equilibrium
Nash equilibrium is named after John Forbes Nash Jr., an American mathematician who won the Nobel Prize for Economics in 1994
Nash equilibrium is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his or her own strategy
Stated simply, Amy and Bill are in Nash equilibrium if Amy is making the best decision she can, taking into account Bill's decision, and Bill is making the best decision he can, taking into account Amy's decision
Likewise, many players are in Nash equilibrium if each one is making the best decision that they can, taking into account the decisions of the others…
Identifying Nash Equilibria : 9 Identifying Nash Equilibria Identifying Nash equlibria in a payoff matrix
The actual mechanics of finding equilibrium cells is as follows:
Find the maximum of a column and check if the second member of the pair is the maximum of the row
If these conditions are met, the cell represents a Nash Equilibrium
Check all columns this way to find all N-E cells
An NxN matrix may have between 0 and NxN pure strategy Nash equilibria Fig. 2 – Example: A Payoff Matrix
Identifying Nash Equilibria – Example : 10 Identifying Nash Equilibria – Example Identifying Nash equlibria - Example
Using this rule, we can very quickly see that the Nash Equlibria cells are (B,A), (A,B), and (C,C).
For the example referred to in Fig. 2, the Nash Equlibria cells are (B,A), (A,B), and (C,C)
Indeed, for cell (B,A) 40 is the maximum of the first column and 25 is the maximum of the second row
For (A,B) 25 is the maximum of the second column and 40 is the maximum of the first row. Same for cell (C,C)
For other cells, either one or both of the duplet members are not the maximum of the corresponding rows and columns Fig. 2 – Example: A Payoff Matrix
Slide 12: 11 The concept of STABILITY, useful in the analysis of many kinds of EQUILIBRIUM, can also
be applied to Nash equilibria.
A Nash equilibrium for a mixed strategy game is stable if a small change (specifically,
an infinitesimal change) in probabilities for one player leads to a situation where
two conditions hold:
the player who did not change, has no better strategy in the new circumstance
2. the player who did change, is now playing with a strictly worse strategy
Slide 13: 12 If these cases are both met, then a player with the small change in his mixed-strategy
will return immediately to the Nash equilibrium. The equilibrium is said to be stable.
If condition one does not hold then the equilibrium is unstable. If only condition one
holds then there are likely to be an infinite number of optimal strategies for the player
who changed.
Stability is crucial in practical applications of Nash equilibria, since the mixed-strategy
of each player is not perfectly known, but has to be inferred from statistical distribution
of his actions in the game. In this case unstable equilibria are very unlikely to arise
in practice, since any minute change in the proportions of each strategy seen will lead to
a change in strategy and the breakdown of the equilibrium.
Decision Making Using GAME THEORY : 13 Decision Making Using GAME THEORY Mix up the strategies to formulate the PAYOFFS
Use any equilibrium such as Nash Equilibrium,Minimax Equilibrium or Dominating Equilibrium.
Use a dominating strategy
Eliminate any dominated strategy
If you have a Dominating strategy, use it : 14 If you have a Dominating strategy, use it Strategy 2 Strategy 1 150 1000 25 Strategy 1 Strategy 2 - 10 You Opponent
Eliminate any Dominated strategy : 15 Eliminate any Dominated strategy Strategy 2 Strategy 1 150 1000 25 Strategy 1 Strategy 2 - 10 You Opponent Strategy 3 -15 160
Game Theory - Applications : 16 Game Theory - Applications Game theory has been used to study a wide variety of human and animal behaviors
It was initially developed in economics to understand a large collection of economic behaviors, including behaviors of firms, markets, and consumers
Economists have used game theory to analyze a wide array of economic phenomena, including auctions, bargaining, duopolies, oligopolies, social network formation, and voting systems
A prototypical paper on game theory in economics begins by presenting a game that is an abstraction of some particular economic situation
One or more solution concepts are chosen, and the author demonstrates which strategy sets in the presented game are equilibria of the appropriate type
Limitations & Problems : 17 Limitations & Problems Assumes players always maximize their outcomes
Some outcomes are difficult to provide a utility
Not all of the payoffs can be quantified
Not applicable to all problems
Slide 19: 18