Game Theory

Question 1

The prisoners dilemma

Answer 1

Two people have been arrested on suspicion of having committed a crime and are being interrogated separately. If they both confess they will both go to jail for 6 years. If both dont confess they will go to jail for 2 years. If one confesses but the other doesnt then the confessor is free and the other gets 10 years.

Question 2

How can a game theoretic situation be represented?

Answer 2

In strategic form

Question 3

Strategic form

Answer 3

A table outlining the players, the strategies and the payoffs

Question 4

Players

Answer 4

the people in the scenario

Question 5

the stratgies

Answer 5

the actions that each player can decide to undertake

Question 6

payoff

Answer 6

what each set of actions results in

Question 7

Matching Pennies

Answer 7

Two Children hold concealed pennies in their hands. The simultaneously reveal their pennies to each other. If their pennies match child 1 gives child 2 their penny, if they dont match then child 2

Question 8

Battle of the Sexes

Answer 8

A man and a woman are trying to decide how to spend a weekend together. Both would prefer to spend the evening together rather than apart, but the man prefers to go to a football game and the woman prefers to go to a concert. The payoffs are represented in utility

Question 9

Game theory

Answer 9

Used to study situations where each person’s payoff depends upon his own actions as well as the actions of other people

Question 10

A strategic form game is characterised by..

Answer 10

1) A set of n players, indexed by i=1,2,…n
2) For every player i, a set of strategies denotedby Si, with si denoting a particular strategy
3) For every player i, a utility function ui(s1,s2…sn)
4) We denote a n-player strategic form game by: G={S1,S2..Sn;u1,u2…un}

Question 11

Dominant strategy equilibrium

Answer 11

If one of the payoff is preferable more so than the other. I.e. the prisoners dilemma, C1 strictly dominates NC1 and C2 strictly dominates NC2. As such the dominant strategy is

Question 12

What is the main problem with the concept of the dominant strategy equilibrium

Answer 12

It does not exist for most games e.g. matching pennies and battle of the sexes

Question 13

Nash Equilibrium

Answer 13

A pair of strategies of a two player strategic form game is a nash equilibrium iff s1bar is the best response to s2bar and simultaneously vice versa

Question 14

Cournot Duopoly

Answer 14

two firms in a market where the price is fixed. The firms choose the quantity to produce.

Question 15

Finding the Nash equilibrium of the Cournot duopoly.

Answer 15

Sub the inverse demand function into Q=q1+q2. Then find the payoff for company 1 and 2. Find the max equation for each company by finding the FOC. Solve for q1 and q2 when q1=q2.

Question 16

Bertrand Duopoly

Answer 16

Similar to the Cournot Duopoly but the firms choose the prices rather than the quatities.

Question 17

What would be the payoff in the Bertrand duopoly

Answer 17

The profit namely total revenue-total costs

Question 18

How to find the Nash equilibrium in the Bertrand Duopoly

Answer 18

Maximise the payoff functions and find the FOC. This finds each firms reaction functions. We then solve for p1 and p2 where p1=p2

Question 19

Games in extensive form

Answer 19

Games in which the the players choose their strategies knowing what other players have chosen in the past.

Question 20

What are the decision the players have in take in extensive form known as?

Answer 20

Actions

Question 21

What is the strategy in the case of extensive form?

Answer 21

Is a specification of an action for each player’s information sets

Question 22

How to convert the extensive form to strategic form

Answer 22

Outline all the strategies for each player. Then set it up in the table form shown in strategic form and write in the payoffs. From there can use the underscoring method to find the Nash equilibrium

Question 23

How to convert strategic form into the extensive form

Answer 23

Start with a node and one of the players and uses the branches to represent their actions from them stems the action of the other player. If the other player does not know the other players actions (e.g. The Prisoners dilemma) then there is a big node.

Question 24

Backwards induction

Answer 24

Way of working out the extensive form starting at the bottom of the tree. For player 2 the past is the past so must do what is best for himself. Player 1 will assume player 2 is rational and so player one will decide the best option for him.

Question 25

Subgame Perfect Nash Equilibrium of an extensive form game

Answer 25

A decision node d and all the sucessor decision nodes such that: i) The information set containing d is a singleton ii) For every decision node d' that is a a sucessor of d and for every decision node d" such that d" is in the same information set as d' must also be a successor of d.

Question 26

Stackelberg Model of Duopoly

Answer 26

Same as the Cournot Duopoly, except that one firm is the 'leader' and makes its output decision first. The other firm, the 'follower', observes the leader's output decision and then makes its own output decision

Question 27

How to solve the Stackelberg Duopoly to find the SPNE

Answer 27

Backwards induction . I.e. Having observes q1 firm two chooses to maximise their quantity. Solve for q2 wrt q1.

Question 28

Wage bargaining

Answer 28

Whereby a union supplies the labour to the firms at a uniform wage. First the union announces the wage rate. Second the firms decide their respective outputs independently. The payoffs for the firms and union are written. Find the SPNE of the Cournot oligopoly subgame given w. Given qa=qb=qc. Then work out what the union maximisation. Solve for w and for each q.

Question 29

Mixed strategies

Answer 29

Allow players to play each of their "pure" strategies with a certain probability rather than choosing to play one particular pure strategy.

Question 30

A mixed strategy

Answer 30

One player's uncertainty about what another player will choose

Question 31

Nah Theorem

Answer 31

In any finite game (involving a finite no. of players and strategies), there exists at least one Nash equilibrium, possibly involving mixed strategies.

Question 32

Bayesian games

Answer 32

Games of incomplete information. So a player knows their own preferences but they are uncertain of the preferences of the other player.

Question 33

A strategic form games of incomplete information is characterised by:

Answer 33

1) A set of n players indexed by i=1,2,3...n 2) for every player i, a set of strategies denoted by Si, with si denoting a particular strategy 3) For every player i, a set of possible types denoted by Ti, with ti denoting a particular type 4) A probability distribution over the possible types 5) for every player i, a utility function u(s1, s2,...sn; ti).

Question 34

A bayesian nash equilibrium

Answer 34

A nash equilibrium of a game of incomplete information

Question 35

Working out a Cournot Duopoly under incomplete informatiom

Answer 35

Firm ones payoff function is common knowledge. Firm 2 knows its payoff function but firm on thinks. There is a high cost associated with probability theta and low cost with prob 1-theta. Work out FOC as if each scenario were true and the FOC for firm 1. This yields three equations and three unknowns. Can then solve...