Game Theory

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Information about Game Theory

Published on October 15, 2008

Author: maddy3

Source: slideshare.net

Game Theory and Strategic Behavior

Developed in 1950s by mathematicians John von Neumann and economist Oskar Morgenstern Designed to evaluate situations where individuals and organizations can have conflicting objectives

Developed in 1950s by mathematicians John von Neumann and economist Oskar Morgenstern

Designed to evaluate situations where individuals and organizations can have conflicting objectives

Any situation with two or more people requiring decision making can be called a game. A game is a description of strategic interaction that includes the constraints on the actions that the players can take and the players’ interests, but does not specify the actions that players do take. A strategy is a course of action taken by one of the participants in a game Payoff is the result or outcome of the strategy Game theory is about choices (finite). While game theory cannot often determine the best possible strategy, it can determine whether there exists one. Game theorists may assume players always act in a way to directly maximize their wins (the Homo economicus model)

Any situation with two or more people requiring decision making can be called a game.

A game is a description of strategic interaction that includes the constraints on the actions that the players can take and the players’ interests, but does not specify the actions that players do take.

A strategy is a course of action taken by one of the participants in a game

Payoff is the result or outcome of the strategy

Game theory is about choices (finite). While game theory cannot often determine the best possible strategy, it can determine whether there exists one.

Game theorists may assume players always act in a way to directly maximize their wins (the Homo economicus model)

Objective – Increase profits by price change Strategies Maintain prices at the present level Increase prices Above matrix shows the outcomes or payoffs that result from each combination of strategies adopted by the two participants in the game No Price Change Price Increase No Price Change Price Increase Firm 2 Firm 1 10, 10 100, -30 -20, 30 140, 35

Objective – Increase profits by price change

Strategies

Maintain prices at the present level

Increase prices

Above matrix shows the outcomes or payoffs that result from each combination of strategies adopted by the two participants in the game

Defined as a set of strategies such that none of the participants in the game can improve their payoff, given the strategies of the other participants. Identify equilibrium conditions where the rates of output allowed the firms to maximize profits and hence no need to change. No price change is an equilibrium because neither firm can benefit by increasing its prices if the other firm does not

Defined as a set of strategies such that none of the participants in the game can improve their payoff, given the strategies of the other participants.

Identify equilibrium conditions where the rates of output allowed the firms to maximize profits and hence no need to change.

No price change is an equilibrium because neither firm can benefit by increasing its prices if the other firm does not

For some games, there may be no Nash equilibrium; continuously switch from one strategy to another There can be more than one equilibrium Firm 2 Firm 1 No Price Change No Price Change Price Increase Price Increase Both firms increasing their price is also a Nash equilibrium 10, 10 100, -30 -20, 30 140, 25

For some games, there may be no Nash equilibrium; continuously switch from one strategy to another

There can be more than one equilibrium

One firm’s best strategy may not depend on the choice made by the other participants in the game Leads to Nash equilibrium because the player will use the dominant strategy and the other will respond with its best alternative Firm 2’s dominant strategy is not to change price regardless of what Firm 1 does

One firm’s best strategy may not depend on the choice made by the other participants in the game

Leads to Nash equilibrium because the player will use the dominant strategy and the other will respond with its best alternative

Firm 2’s dominant strategy is not to change price regardless of what Firm 1 does

An alternative that yields a lower payoff than some other strategies a strategy is dominated if it is always better to play some other strategy, regardless of what opponents may do It simplifies the game because they are options available to players which may be safely discarded as a result of being strictly inferior to other options.

An alternative that yields a lower payoff than some other strategies

a strategy is dominated if it is always better to play some other strategy, regardless of what opponents may do

It simplifies the game because they are options available to players which may be safely discarded as a result of being strictly inferior to other options.

A strategy s¡ in set S is strictly dominated for player i if there exists another strategy, s¡’ in S such that, Π i(s¡’) > Π i(s¡) In this case, we say that s¡’ strictly dominates s¡ In the previous example for Firm 2 no price change is a dominant strategy and price change is a dominated strategy

A strategy s¡ in set S is strictly dominated for

player i if there exists another strategy, s¡’ in S such that,

Π i(s¡’) > Π i(s¡)

In this case, we say that s¡’ strictly dominates s¡

In the previous example for Firm 2 no price change is a dominant strategy and price change is a dominated strategy

Highly competitive situations (oligopoly) Risk-averse strategy – worst possible outcome is as beneficial as possible, regardless of other players Select option that maximizes the minimum possible profit

Highly competitive situations (oligopoly)

Risk-averse strategy – worst possible outcome is as beneficial as possible, regardless of other players

Select option that maximizes the minimum possible profit

Each firm first determines the minimum profit that could result from each strategy Second, selects the maximum of the minimums Hence, neither firm should introduce a new product because guaranteed a profit of at least $3 million Maximin outcome not Nash equilibrium- loss avoidance rather than profit maximization Firm 1 Firm 2 Firm 2 Minimum Firm 1 Minimum New Product No New Product No New Product New Product 3 2 2 3 4, 4 3, 6 6, 3 2, 2

Each firm first determines the minimum profit that could result from each strategy

Second, selects the maximum of the minimums

Hence, neither firm should introduce a new product because guaranteed a profit of at least $3 million

Maximin outcome not Nash equilibrium- loss avoidance rather than profit maximization

Pure strategy – Each participant selects one course of action Mixed strategy requires randomly mixing different alternatives Every finite game will have at least one equilibrium

Pure strategy – Each participant selects one course of action

Mixed strategy requires randomly mixing different alternatives

Every finite game will have at least one equilibrium

Non cooperative games Cooperative games Repeated games Sequential games

Non cooperative games

Cooperative games

Repeated games

Sequential games

Not possible to negotiate with other participants Because the two participants are interrogated separately, they have no idea whether the other person will confess or not

Not possible to negotiate with other participants

Because the two participants are interrogated separately, they have no idea whether the other person will confess or not

Possibility of negotiations between participants for a particular strategy If prisoners jointly decide on not confessing, they would avoid spending any time in jail Such games are a way to avoid prisoner’s dilemma

Possibility of negotiations between participants for a particular strategy

If prisoners jointly decide on not confessing, they would avoid spending any time in jail

Such games are a way to avoid prisoner’s dilemma

Yet another way to escape prisoner’s dilemma If exercise is repeated multiple times, reactions become predictable Acc. to eg in PD, both firms select high advertising & capture max. profit But, if this exercise is repeated, outcomes may change Advantage becomes temporary Winning strategy- ‘tit for tat’

Yet another way to escape prisoner’s dilemma

If exercise is repeated multiple times, reactions become predictable

Acc. to eg in PD, both firms select high advertising & capture max. profit

But, if this exercise is repeated, outcomes may change

Advantage becomes temporary

Winning strategy- ‘tit for tat’

Infinitely Repeated Game Co-operative behaviour is a rational response to a tit for tat strategy Finite Number of Repetitions Strategise to take action in the last period of time in order to have a long term effect

Infinitely Repeated Game

Co-operative behaviour is a rational response to a tit for tat strategy

Finite Number of Repetitions

Strategise to take action in the last period of time in order to have a long term effect

One player acts first & then the other responds 2 firms contemplating the introduction of an identical product in the market 1 st firm- develop brand loyalties, associate product with the firm in minds of consumers Thus, first mover advantage

One player acts first & then the other responds

2 firms contemplating the introduction of an identical product in the market

1 st firm- develop brand loyalties, associate product with the firm in minds of consumers

Thus, first mover advantage

Assume firms use maximum criterion, so neither should introduce a new product and earn $2 mn each Firm 1 introduces a new product, firm 2 will still decide to stay out because right now it is losing $5 mn, opposed to $7 mn otherwise. Firm 2 No new product Introduce new product Firm 1 No new product $2, $2 $-5, $10 Introduce new product $10, $-5 $-7, $-7

Assume firms use maximum criterion, so neither should introduce a new product and earn $2 mn each

Firm 1 introduces a new product, firm 2 will still decide to stay out because right now it is losing $5 mn, opposed to $7 mn otherwise.

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