Game Theory - Static games

Question 1

How do you find and graph mixed strategies?

Answer 1

To find Best Response function:

1. Find the expected payoff for each strategy for player 1
2. Find the probability where the player is indifferent between each strategy àset the payoff’s equal to each other and solve for p
3. Diagram best response functions
4. Repeat for player 2
5. Where best response functions meet = Nash equilibria

Question 2

What is a mixed strategy?

Answer 2

A player has M pure strategies, . A mixed strategy for this player is a probability disruption over their pure strategies – a probability vector (p₁,p₂,…,p_M), with p_k >=0, for k=1,..,M, and the sum of probailties sum to 1 .

Question 3

Why are economist sceptical of mized strategies?

Answer 3

Because we are bad at randomising, and it seems unreasonable to assume players able to correctly guess the exact probabilities that are being used by the other players .

Question 4

Define best response

Answer 4

Best response= the strategy that yields highest payoff given strategies of other players.

Player i’s best response to is , if

Question 5

Define dominant strategy

Answer 5

Dominant strategy = a strategy that is better than another regardless of the strategies of player’s opponents.

Strategy, is a dominant strategy for player i if it is a best response to all

Question 6

What is a strictly dominated stategy?

Answer 6

Strictly dominated strategy, if player I could commit to never playing it.

Strategy is a strictly dominated strategy if there exists some such that for any

Question 7

Define pareto efficent ourcome, in terms of game thoery

Answer 7

Pareto efficient= it is impossible to make one player better off without making another worse off

is Pareto efficient if there does no exist some such that for every i, with strict inequality for some i.

Question 8

Define nash equilbrium (general defintion including mixed strategies)

Answer 8

Nash equilibrium= in the two-player normal-form game G= , the mixed strategies are a Nash equilibrium if each players mixed strategy is a best response to the other player’s mixed strategy.

For the pair of mixed strategies to be Nash equilibrium:

must satisfy: for every distribution over

must satisfy: for every distribution over

Question 9

What is a correlated equilbrium?

Answer 9

An equilbrium when players enter pre-play arrangments in order to achieve higher pay-off equilbrliums - breaks the assumption that players cannot enter pre-play arrangements

Question 10

1,2

S

GS

S

0,0

-3,2

GS

2,-3

-6,-6

Find the mixed strategy equlibrium for this game of chicken, and show how a correlated equilbrium can achieve higher pay offs (mathmatically and graphically)

Answer 10

Mixed strategy Nash equilibrium: mix probabilities where is the probability player 1 plays S, and is the probability player 2 plays S - with expected payoff =

Payoffs can be changes by players entering in to pre-play arrangements such as:

1. A coin toss - players agree to observe a coin toss and

play (S,GS) if “heads” and (GS,S) if “tails”. Expected payoffs = This is a convex combination of 2 pure-strategy Nash equilibria. Players end up in an Nash equilibria (with a higher payoff) in the convex hull of the set of Nash equilibria of the biomatrix game

1,2

S

GS

S

GS

0

2. A mechanism that randomly selects 1 cells in the game matrix with the following probabilities. When a cell is selected, each player is told by ‘umpire’ to play corresponding to that cell, but not what the other player is told and information is not public (so to not incentivise plays to cheat). If player 1 receives signal “Play S’ the they know that plays 2 has received the signal S with probability and GS with probability àpayoff = , by deviating Play 1 gets payoff= , so player 2 does not wish to deviate. If player 1 received the signal ‘Play GS”, the they know player 2 received the signal S. This is true for player 2 as well, therefore it is an equilibrium

Question 11

Describe a game of Hawk-Dove, and what conditions it becomes a game of chicken or a prisoners dilemma.

Answer 11

The game of chicken is also know is Hawk-Dove, where players (which could be animals) contest a scare resource. Each player can choose to fight, aka. play hawk, H, or not to fight, aka, play dove, D. The benefit of the resource is V, and the cost of a fight is C.

1,2

H

D

H

V,0

D

0,V

For this to be a game of chicken, V>C, otherwise it is a prisoners dilemma.

Question 12

Describe a game of battle of the sexes

Answer 12

Common objective, but have different preferences on how to do that and are unable to communicate.

No dominant strategy, and 2 Nash equilibriums

e.g. a couple are determining whether to go to football or the opera, and have different preferences, but would rather go to the one the prefer less with their spouse than to the one they want alone.

1,2

Football

Opera

Football

3,1

0,0

Opera

0,0

1,3

Question 13

What is maxminimisation? (in words and maths)

Answer 13

Maxminimisation maximises player i’s expected pay-off under the assumption that the other play j will act in a way that would minimise player i’s payoff.

Definition: A maxminimising mixed strategy for player i is a mixed strategy that solves the problem: Where is player i’s expected payoff to the mixed strategy profiles.

Question 14

How do you solve for a maxminimsier stategy? (mathametically and graphically)

Answer 14

1,2

L

R

U

2,-2

-1,1

D

-1,1

1,-1

Example:

Let ) be player 1’s mixed strategy, where is the probability Player 1 assigns to U.

The expected payoff to P1 when P2 plays L is:

The expected payoff to P1 when P2 plays R is

The lower of the 2 lines (in bold) indicates player 1’s lowest payoff, aka. = Therefore the maxminimising strategy for player 1 is

Question 15

What is the realtionship between maxminisation and nash equilbrium?

Answer 15

A player’s expected payoff in a mixed strategy Nash equilibrium strategic game is at least equal to their maxminimised payoff.

Question 16

Define a strictly competative game

Answer 16

Players have exactly opposite ranking over the outcomes of the games – they are diametrically opposed

A strategic game is a strictly competitive if it has 2 players &

Question 17

What is the requirments for a NE in a strictly competative game?

Answer 17

Each player’s strategy is a maxminimiser.

Question 18

Which of these games are strictly competative?

Answer 18

Prisoners dilemma, battle of the sexes and chicken are NOT strictly competitive games, as players would prefer a specific outcome to at least one of the outcomes.

Matching pennies is an example of a strictly competitive game.

Question 19

What is a game of incomplete infomation? And what do players optimise based off of?

Answer 19

Given a players type, optimise player’s strategy based on their type and their opponent possible strategies. Players respond to plans, rather than actions, as type is unknown.

Question 20

Define a strategy in a game of incomplete infomation

Answer 20

Strategies are plans of action, with incomplete info it is not possible to let plans be made contingent on other players moves. They can be made contingent on the players own types (although not on the player’s types)

Strategy describes what player i will do in a static game if she is of type tⁱ

Question 21

Define a bayes-nash equilbrium

Answer 21

Bayes-Nash equilibrium: A profile of strategies is a Bayes-Nash Equilibrium if, for every i, sⁱis a best response to

Question 22

How do you solve for a bayes-nash equilbrium in a prisoners dilemma with 1-sided info assymetry?

Answer 22

1. Draw out table that summaries payoffs, and the type probability distribution
2. As player 1 type is known to them, calculate conditional probabilities of the other players type
3. Fix player 2 strategy and calculate expected pay off of each of player 1’s strategies for each of player 2’s types
4. Show that a specific strategy is a BR to player 2’s strategy - Bayes Nash equilibrium

Question 23

Find the optimum quantities in a cournot competition of one-sided info asymmetry (mathmatically)

Where:

Firm 2’s MC is unknown to Firm 1, where as Firm 2 has complete info

c¹= firm 1 costs – which is common knowledge

Firm 2’s costs depends up its type, .

If firm 2 is its , and if , and

Answer 23

Firm 2’s MC is unknown to Firm 1, where as Firm 2 has complete info

c¹= firm 1 costs – which is common knowledge

Firm 2’s costs depends up its type, .

If firm 2 is its , and if , and

1. Start by analysing output decision of informed firm – firm 2 – and solve for both of its types, with denoting firm 2’s conjecture about firm 1 output àsolving for each type we get: and
2. Look at firm 1 (uninformed about firm 2) -
3. Calculate - As probability of L/H firms are equal -
4. Solve for
5. Solve for and
6. Graph àdotted line = expected best response for firm 2, and firm 1 uses this to make decisions - assumption is risk neutral (wont have risk loving or risk adverse as too complex)

Question 24

Find the optimum quantities in a cournot competition of one-sided info asymmetry (graphically)

Where:

Firm 2’s MC is unknown to Firm 1, where as Firm 2 has complete info

c¹= firm 1 costs – which is common knowledge

Firm 2’s costs depends up its type, .

If firm 2 is its , and if , and

Question 25

Find the optimum quantities in a cournot competition of two-sided info asymmetry (mathmatically)

Where:

Firm’s MC is unknown to other firm

Firm 2’s costs depends up its type, .

If firm 2 is its , and if , and

Answer 25

We assume that the two types of firm occur with equal probability

Since firms are assumed to be symmetric, we can simply let for t

Profit maximising, we get: and

Solving for

Solving for and : and

Graphically àboth have 2 lines, and Nash equilibrium where average = average

Question 26

Find the nash equilbrium for

Question 27

How does an English auction work?

Answer 27

ascending bid auction – starts low and goes higher until 1 person is left

Question 28

How does a Dutch auction work?

Answer 28

descending bid auction – starts high and goes lower until 1 person bids

Question 29

How does a seal-bid auction work (1st price and 2nd price)?

Answer 29

highest bidder is given the object, the amount they pay varies:1st price auction – bidder has to pay her bid 2nd price action – bidder has to pay the second highest bid

Question 30

What is a truthful bid?

Answer 30

bid equal to players valuation

Question 31

Show that in 2nd price seal-bid auction, it is the best response to submit a truthful bid

Answer 31

1. Suppose player 1 considers a different strategy,, and

If

True valuation

Vs.

Reduced bid

=

>

>

0

0

=

0

=

0

no incentive to bid lower than your true valuation

2. Suppose player 1 considers a different strategy,, and

If

True valuation

vs.

Increased bid

=

0

=

0

>

0

(as pay more than your valuation)

0

>

<0

0

=

0

no incentive to bid higher than your true valuation

no incentive to deviate for your valuation àweakly dominate strategy is to submit truthful bid – by symmetry all players play this – so there is a Bayes Nash equilibrium

Question 32

Explain the expected surplus of a 2 person 2nd price auction where the bidders can vary in type, , each type is equally likely + sellers want to maximise the expected sale price.

Answer 32

Its a dominant strategy to play truthful bids.

If player is of type gets the object under 2 circumstances, (a) if the other player is , or (b) the other player is type, then player 1 has a 50% chance of winning.

If player is of type gets the if the object if player is with 50% probability

Probability of winning:

Expected payments:

Expected surplus:

Question 33

Explain why in a 1st price acution truthful bidding is NOT a dominant strategy

Answer 33

If they win and have bid truthfully surplus = 0, but if reduce bid surplus = +ve. They only don’t have regret if they win the object by a negligible amount above their opponents bid there is no pure strategy best response - mixed strategy

Question 34

Explain the players dominant strategy for a 2 player 1st price acution.

Where bidders can vary in type, , each type is equally likely + sellers want to maximise the expected sale price.

Answer 34

(Let p be a bid and q be a bid)

For - NO - If bids truthfully, they could do better against a -type but reducing their bid. In Bayes Nash equilibrium

For - YES - In Bayes Nash equilibrium -type bids , because if and the -player plays against another -type they win by bidding slightly higher than q, the other -type will use the same reasoning until they both bid .

Question 35

Proove that there is a symmetric Bayes-Nash equilibrium of the first-price auction in which -type bidder bids her valuation and the -type plays a mixed strategy that assigns a probability to all bids between (+ some negligible amount) and . For any bid between those extremes the probability that the opponents bid will be less than s is given by .

Answer 35

For : if played mixed strategy, incentive to increase lower bound of bids to beat opponent, hence is the only best response

For : minimum bid must be + some negligible amount, as if it is lower they could lose against a -type.

Question 36

Show that there is no other Bayes-Nash equilbrium in two player first-price auction other than -type bidder bids her valuation and the -type plays a mixed strategy that assigns a probability to all bids between (+ some negligible amount) and . For any bid between those extremes the probability that the opponents bid will be less than s is given by .

Answer 36

For : if played mixed strategy, incentive to increase lower bound of bids to beat opponent, hence is the only best response

For : minimum bid must be + some negligible amount, as if it is lower they could lose against a -type.

If they bid the probability the other player if bids that is so small it is negligible only win if other player is surplus =

For mixed strategy – all responses must give the same surplus all surplus =

From this cumulative probability function you can work out probability distribution (not examinable).

upper-bound is as would win with 100% probability so no incentive to bid higher than this.

Thus, the this is the only Bayes Nash equilibrium.

Question 37

Show that there is revenue equivilance between two player first and second auctions, given:

The net surplus for the is 0 and for the it is àsame as 2nd price auction.

The probability of winning in the two auctions is the same for each type: probability that the wins is 1/4, and the probability that the wins is 3/4

Answer 37

The net surplus for the is 0 and for the it is àsame as 2nd price auction.

The probability of winning in the two auctions is the same for each type: probability that the wins is 1/4, and the probability that the wins is 3/4

àà

Which is the same as the second price auction