Microeconomics 10: Game theory and Oligopoly

Question 1

Describe the simplest way to illustrate Game Theory

Answer 1

• The simplest way to illustrate a game is with a payoff matrix – this can be used when we have two players
  choosing from a finite number of strategies (i.e. they
  do not have a continuous choice).
• In the simplest case, we have two strategies for each
  player.
• We normally have the first player choosing a row,
  and the second choosing a column.

Question 2

Using an example, describe ‘dominant strategy equilibrium’ in Game Theory

Answer 2

A simple example:

                          B chooses Left    B chooses right A chooses Top          1, 2                             0,1  A chooses                  2,1                             1, 0 bottom

• The numbers in each cell represent the payoffs to
  each player, starting with the row player (A in this
  case), then the column player (B).
• A higher number is better for the player than a lower number.
• If B chooses Left, A prefers Bottom (2 > 1);
• If B chooses Right, A also prefers Bottom (1 > 0).
• So A will always play Bottom – we call this a
  dominant strategy for A
• Similarly for B, Left is always better than Right (2 > 1 when A chooses Top; 1 > 0 when A chooses Bottom), so this is B’s dominant strategy.
• The equilibrium is (Bottom, Left) with payoffs of (2,
  1).
• We call this a dominant strategy equilibrium

Question 3

Describe Nash Equilibrium

Answer 3

• Nash equilibrium is based on the concept of best
  response:
  – An action is a best response to another action if it is the
  best the player can do, given what the other player is
  doing
  – This means there is no incentive to change action.
• Note that it is not necessarily true that the same
  action will be a best response to every action by
  another player – that means we may not have a
  dominant strategy, but could have different best
  responses in different situations.

Question 4

What’s Game Theory?

Answer 4

Game theory is the analysis of strategic interaction. Within Economics, it
can cover interactions between individuals, firms or governments.

Question 5

Describe whether Nash equilibrium is necessarily unique

Answer 5

• For a Nash equilibrium to exist, we need both players
  simultaneously to be playing a best response to each
  other’s actions:
  – When neither player can gain from unilaterally changing their
  action. it proves that Nash equilbrium is not necessarily unique

Question 6

Describe & explain, with an example including a payoff matrix, when there’s no pure strategy Nash equilibrium

Answer 6

B chooses Left B chooses right
A chooses Top 1, 0 0,1
A chooses 0,1 1, 0
bottom

• In this game, whichever player has a payoff of zero
  would always gain by changing their strategy.
• For example, if we start at (Top, Left), B would switch
  to playing Right; from there, A would switch to
  playing Bottom; and so on
  We say in this case there is no pure strategy Nash
  equilibrium – that is, no equilibrium where each
  player plays a given strategy with certainty.
• However there would still be a mixed strategy
  equilibrium where each player randomises over their
  two choices, playing each with a certain probability.

Question 7

Describe & explain ‘The Prisoner’s Dilemma’

Answer 7

• Two prisoners are being interviewed separately
  about a crime they have committed.
  – There is little other evidence available to the police, but
  without a confession they can prove that a minor offence
  was committed, for which each prisoner would receive a 1-
  month sentence.
  – If one prisoner confesses, while placing most of the blame
  on the other, the prisoner who confesses will be released
  while the other prisoner will receive a 6-month sentence.
  – If both prisoners confess, each will receive a 3-month
  sentence. Payoffs are minus the length of the sentence

                              B confess    B deny A confess                   1, 0                 0,1  A deny                        0,1                 1, 0

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