Game Theory

Question 1

What is a strategy?

Answer 1

A decision rule that describes the actions a player will take at each decision point

Question 2

What is normal form?

Answer 2

A representation of a game indicating the players, their possible strategies, and the payoffs resulting from alternative strategies

Question 3

What is normal form convention?

Answer 3

Convention: row player = 1, column player = 2

Question 4

What is a dominant strategy?

Answer 4

Results in the highest payoff to a player regardless of the opponents action

Question 5

What is a secure strategy?

Answer 5

Guarantees the highest payoff given the worst possible scenario

Question 6

What is strictly dominating?

Answer 6

A strategy of player i is strictly dominated if there is another strategy available to player i which yields strictly higher profits regardless of the strategies chosen by other players

Question 7

What is Nash eqm?

Answer 7

• A condition describing a set of strategies in which no player can improve their payoff by unilaterally changing their own strategy, given the other players’ strategies
• i.e. once we get here, no one has incentive to change

Question 8

What is a best response?

Answer 8

For any given strategy of the other player, the best response for player i prescribes the best strategy(ies) which give them the highest payoff

Question 9

What is a Nash eqm to best strategies?

Answer 9

A Nash equilibrium is a strategy profile such that every player’s strategy is a best response to the strategies of all the other players

Question 10

What is true of the nash eqm for duopoly games?

Answer 10

• Nash equilibrium: both players charge ‘low price’
• Payoffs associated with the Nash eq is inferior from the firms’ viewpoint compared to both ‘agreeing’ to charge ‘high price’: hence the prisoners dilemma

Question 11

What are the caveats of the Nash eqm?

Answer 11

• Nash eqm does not always give a unique prediction
• There are games without pure strategy Nash Eqm
  
  • Finite games always has at least one Nash eqm if considering
    mixed strategy eqm as well
• Nash eqm foes not always generate the ‘best possible’ outcome
  
  • prisoners’ dilemma
  • Role of commitment and communication?
  • Trust, reputation and repeated game?

Question 12

What is a pure strategy?

Answer 12

Pure strategy: strategy in which a player plays a certain action with probability 1: Monitor or Don’t Monitor

Question 13

What is a mixed strategy?

Answer 13

Mixed strategy: strategy in which a player makes a random choice among two or more possible actions, based on a set of chosen probabilities: 50% monitor + 50% don’t monitor

Question 14

What does a Nash eqm in mixed strategy refer to?

Answer 14

• Each agent chooses the optimal frequency with which to play their strategies
• Given the frequency choices of the other agent

Question 15

What does mixed strategy imply about the Nash eqms?

Answer 15

If considering mixed strategy as well, every game with a finite number of player and a finite number of actions has at least one Nash eqm

Question 16

How is the Nash eqm for mixed strategy determined?

Answer 16

• Still have to satisfy: any given player is playing his best response given other player’s strategies, and no player can increase his payoff by unilateral deviation
• For mixed strategy eqm with strictly positive probability on more than 2 strategies, the player must be indifferent to the pure strategies they are mixing over given the other players’ strategies

Question 17

What is true of inequality in mixed strategy and what does this imply?

Answer 17

• If EV of monitor > EV of don’t monitor, Manager will monitor with probability 1
• For manager to play mixed strategy, worker must be randomising over {work, don’t work} with probability (1/2,1/2)

Question 18

What are multistage games?

Answer 18

• Player make sequential, rather than simultaneous decisions
• Represented by an extensive-form game
  
  • A representation of a game that summarises the players, the information available to them at each stage, the strategies available to them, the sequence of moves, and the payoffs resulting form alternative strategies

Question 19

What is the difference between strategies and actions?

Answer 19

• A strategy in a dynamic game is a complete contingent plan
  
  • i.e. telling us what player 2 would do in both decision nodes
  • Player 1 has 2 strategies, player 2 has 4

Question 20

What is a subgame?

Answer 20

• A smaller game ‘embedded’ in the complete game
• Starting from some point in the original game
• Includes all subsequent choices that must be made if the players actually reached that point in the game

Question 21

What is a subgame perfect nash eqm?

Answer 21

• A strategy profile is a subgame perfect Nash eqm if the strategies are a Nash eqm in every subgame
  
  • We require that the player play’s Nash eqm strategies in every subgame
• A stronger requirement than Nash eqm
  
  • The player anticipates that its opponent would behave optimally
  • No non credible threat all outcomes must be the individual players Nash Eqm
• Since the game itself is a subgame, a subgame perfect Nash eqm if also a Nash eqm

Question 22

What is backward induction?

Answer 22

• In a finite game of perfect information, the subgame perfect Nash eqm can be found through backward inducation
• Identify the optimal moves for the smallest subgame
• When making decisions in each subgame, players recognise that all the players will behave optimally further down the tree
• Move backwards until the Nash moves for every possible subgame have been found
• The collection of moves for every player at every subgame constitutes the subgame perfect Nash eqm strategies

Question 23

When a firm earns the same profit π, in each period over an infinite time horizon, the PV of the profit stream:

Answer 23

PVfirm = ((1+i)/i)π

Question 24

What is ((1+i)/i)π ?

Answer 24

When a firm earns the same profit π, in each period over an infinite time horizon, the PV of the profit stream

Question 25

What is the equilibrium in a finite game?

Answer 25

If the stage game has a unique Nash eqm, then for any finite number of repetitions, the repeated game has a unique subgame perfect equilibrium: Nash Eqm is played in every repeated stage

Question 26

What happens in infinitely repeating games?

Answer 26

• When this game is repeatedly played for infinite periods, it may be possible for firms to collude by using trigger strategies
  
  • A strategy that is contingent on the past play of a game and in which some particular past action ‘triggers’ a different actions by a player

Question 27

What is the equilibrium in an infinitely repeating game?

Answer 27

Both firms charge the high price, provided neither of them has ever cheated in the past. if one charges low price, the other will punish the deviator by charging low price forever after

Question 28

How is collusion sustained?

Answer 28

• In any given period, cheating gets πcheat, charging high price gets (πcoop(1+i))/i
• High prices can be sustained as an eqm if (πcoop(1+i))/i > πcheat i.e. if i is small enough
• TOTAL πcheat = πcheat + πcompete/i

Question 29

What makes sustaining collusions with trigger strategies more likely

Answer 29

• Know who their rivals are, so they know who to punish
  
  • n x (n-1) firms need to be monitored constantly
• Know who their rivals customers are, so they can ‘steal’ them with lower prices
• Know when their rivals deviate
• Be able to to successfully punish rival

Question 30

What occurs in a finitely repeating game with a known final period?

Answer 30

• When this game is repeated some known, finite number of times and there is only one Nash eqm, then collusion cannot work
• The only eqm is the single-shot, simultaneous move Nash eqm (low,low)

Question 31

What occurs in a finitely repeating game with an unknown final period?

Answer 31

• Suppose the probability that the game will end in any given period is ø, where 0 ≤ ø ≤ 1
• An uncertain final period mirrors the analysis of infinitely repeated games. Use the same trigger strategy
• No incentive to cheat on the collusive outcome provided:
  
  • πcheat ≤(π/ø = πcoop)
• i.e. depends on a small enough ø

Question 32

When working out whether or not collusion is sustainable, what is the collusive profit (πcollude(1+i)/i) compared to?

Answer 32

• TOTAL πcheat = πcheat + (πcompetition/i)

- πcompetition/i represents the PV of future cournot eqm profits

Question 33

What REALLY is the inequality to satisfy to determine whether collusion can be sustained or not?

Answer 33

(πcollude(1+i)/i) > (πcheat + (πcompete/i))

Question 34

How can a mixed strategy (over A and B) be made to strictly dominate another (C), against strategies X and Y for the other player?

Answer 34

P(π=(a,x))) + (1-P(π=(b,x))) > (π=(c,x))
AND
P(π=(a,z))) + (1-P(π=(b,z))) > (π=(c,z))

Question 35

What is essential about sequential games?

Answer 35

Remember threats! P2 will threaten to try to incentivise P1 to play the strategy which P2 wants

Question 36

What is essential about a games Nash eqm’s?

Answer 36

Includes incredible threats

Question 37

What is πcollude when there is a ø chance of the period ending?

Answer 37

πcollude/ø