Game Theory

1
Game Theory

• Our discussion comes from
• Bierman, H. Scott and Luis Fernandez. 1998. Game
  Theory with Economic Applications. 2nd ed.
  Addison-Wesley.
• Game Theory the study of how individuals make
  decisions when they are aware that their actions
  affect each other and when each individual takes
  this into account.
• History Introduced in 1944 by John von Neumann
  and Oskar Morgenstern in The Theory of Games and
  Economic Behavior.
• The work of von Neuman and Morgenstern was
  expanded upon by John Nash.

2
Introduction to Game Theory

• A game is a situation in which a decision-maker
  must take into account the actions of other
  decision-makers. Interdependency between
  decision-makers is the essence of a game.
• In games people must make strategic decisions.
  Strategic decisions are decision that have
  implications for other people.
• Game Elements
• 1. Set of Players.
• 2. Order of Play.
• 3. Description of the information available to
  any player at any point during the game.
• 4. Set of actions available to each player when
  making a decision.
• 5. Outcomes that result from every possible
  sequence of actions by the players.
• 6. A payoff from the outcomes.
• 7. Strategic situations with the above elements
  is considered to be well defined.

3
Cooperative and Non-Cooperative Games

• Non-Cooperative Games are games in which players
  cannot enter binding agreements with each other
  before the play of the game.
• Cooperative Games are games in which players can
  enter binding agreements with each other before
  the play of the game.
• In class we only review non-cooperative games.

4
The Players

• Players in a Game
• 1. Players are decision-makers.
• 2. Nature is a special type of player. Nature
  chooses
• actions according to fixed probabilities.
• 3. Strategic players are assumed to be rational
  decision-makers.

5
Three Basic Assumptions

• Assumption 1 Decision-makers are rational.
• Rational behavior Decision-makers are assumed to
  make their choices according to internally
  consistent criteria.
• Assumption 2 The rationality of all players is
  common knowledge.
• Common knowledge A fact in a game is said to
  be common knowledge if every player know it, and
  every player knows that every other player knows
  it, etc.
• Assumption 3 The complete description of the
  game (players, actions, strategies, order of
  play, information, and payoffs) is also common
  knowledge.
• Time permitting, we may examine games where this
  last assumption is relaxed.
• With these assumptions in hand, we can look at
  the order of play. Before we discuss this issue,
  we must first review decision theory.

6
Order of Play

• Order of Play is the sequence in which decisions
  are made.
• Sequential-move game is a game where players make
  decisions in a sequence, one after another like
  chess (which we deal with later).
• Simultaneous-move game is a game where players
  make their decisions at the same time, such as
  most sports.
• To understand how moves are made, we must briefly
  discuss the issue of information.

7
Information

• Information any observation or knowledge that
  would lead someone to reevaluate her/his
  probability assessments.
• If information has value, the following must be
  true
• 1. the information must alter the decision
  makers optimal action at some decision node.
• 2. the information must be revealed to the
  decision maker before the critical decision
  node is reached.
• Perfect recall A game has perfect recall if no
  player forgets any information she/he once knew,
  and all players know the actions they previously
  took.
• Perfect information Every player as every
  decision node knows the actions taken previously
  by every other player (including Nature).
• Imperfect information Players do not have
  perfect information.

8
Actions, Strategies, and Payoffs

• Actions The set of choices available at each
  decision node in a game.
• Pure strategy a rule that tells the player what
  action to take at each of her information sets in
  the game.
• Mixed strategy when players can choose randomly
  between the actions available to them at every
  information set.
• Example Play calling in sports is a mixed
  strategy.
• Payoffs, for our purposes, consist of either
  profits to firms, or income to individuals.
  Payoffs can also be characterized in terms of
  utility.

9
Solving Games Nash Equilibrium

• Solution Concept a methodology for predicting
  player behavior.
• Nash Equilibrium - a collection of strategies one
  for each player, such that every player's
  strategy is optimal given that the other players
  use their equilibrium strategy.

10
Dominant and Dominated Strategies

• Payoff matrix a matrix that displays the
  payoffs to each player for every possible
  combination of strategies the players could
  choose.
• Dominant Strategy a strategy that is always
  strictly better than every other strategy for
  that player regardless of the strategies chosen
  by the other players.
• Dominated Strategy a strategy that is always
  strictly worse than some other strategy for that
  player regardless of the strategies chosen by the
  other players.

11
Weakly Dominate Strategies

• Weakly dominant strategy - a strategy that is
  always equal to or better than every other
  strategy for that player regardless of the
  strategies chosen by the other players.
• Weakly Dominated Strategy a strategy that is
  always equal to or worse than some other strategy
  for that player regardless of the strategies
  chosen by the other players.

12
Prisoners Dilemma

• Scenario Two people are arrested for a crime
• The elements of the game
• The players Prisoner One, Prisoner Two
• The strategies Confess, Dont Confess
• The payoffs
• Are on the following slide (payoffs read 1,2)
  

13
Prisoners Dilemma, cont.

• Prisoner 2
• Confess Dont Confess
• Confess 2 years, 2 years 0 years, 10 years
• Prisoner 1
• Dont Confess 10 year, 0 years 5 years, 5
  years
• Dominant strategy equilibrium In this game, the
  dominant strategy for each prisoner is to
  confess. So the outcome of the game is that they
  each get two years.
• This illustrates the prisoners dilemma games in
  which the equilibrium of the game is not the
  outcome the players would choose if they could
  perfectly cooperate.

14
The Advertising Game

• Scenario Two firms are determining how much to
  advertise.
• The elements of the game
• The players Firm 1, Firm 2
• The strategies
• High advertising, low advertising

15
Advertising Game, Cont.

• The payoffs Are as follows (payoffs read 1,2)
• Firm 2
• High Low
• High 40,40 100, 10
• Firm 1
• Low 10, 100 60,60
• Dominant strategy equilibrium In this game, the
  dominant strategy for firm 1 and firm 2 is high.
  So the outcome of the game is 40,40.
• Again, this is an example of the prisoners
  dilemma. The equilibrium of the game is not the
  outcome the players would choose if they could
  cooperate.

16
More Prisoner Dilemmas

• Industrial Organization Examples
• Cruise Ship Lines and the move towards glorious
  excess. Royal Caribbean offers a cruise with an
  18 hole miniature golf course. Princess Cruises
  has a ship with three lounges, a wedding chapel,
  and a virtual reality theater.
• Owners of professional sports teams and the
  bidding on professional athletes.
• Non-IO Examples
• Politicians and spending on campaigns.
• Worker effort in teams. The incentive exists to
  shirk, a strategy that if followed by all
  workers, reduces the productivity of the team.
  More on shirking later.

17
Iterated Dominant Strategies

• What if a dominant strategy does not exist?
• We can still solve the game by iterating towards
  a solution.
• The solution is reached by eliminating all
  strategies that are strictly dominated.

18
Example of Iterated Dominance

• Down is Firm 1, Across is Firm 2

19
Alternative Solution Strategies

• Nash Equilibrium - a strategy combination in
  which no player has an incentive to change his
  strategy, holding constant the strategies of the
  other players.
• Joint Profit Maximization This is the objective
  of a cartel.
• Cut-Throat A strategy where one seeks to
  minimize the return to her/his opponent.
• How does the previous game change when we change
  the objectives of the players?
• This is one of the advantages of game theory. We
  do not have to assume profit maximization. We
  still need to be able to identify the objectives
  of the players.

20
A Lack of Dominance

• Down is Player 1, Across is Player 2

21
A Lack of Dominance, cont.

• Given these payoffs, is there a dominant or
  dominated strategy?
• If 1 chooses A, 2 will choose C
• If 1 chooses B, 2 will choose B
• If 1 chooses C, 2 will choose A
• Likewise
• If 2 chooses A, 1 will choose A
• If 2 chooses B, 1 will choose B
• If 2 chooses C, 1 will choose C
• Therefore, no dominant or dominated strategy
  exists. Is there a Nash equilibrium?
• What if player 1 chose C, and player 2 chose A,
  is this a Nash Equilibrium?
• No, if player 2 chose A, player 1 would want A.
• Only when both choose B, or both happy with the
  choice, therefore this is a Nash equilibrium.

22
Mixed Strategy

• Pure Strategy is a rule that tells the player
  what action to take at each information set in
  the game.
• Mixed strategy allows players to choose randomly
  between the actions available to the player at
  every information set. Thus a player consists of
  a probability distribution over the set of pure
  strategies.
• Examples of mixed strategy games
• Play calling in sports
• To shirk or not to shirk

23
The Shirking Game

• Scenario A worker is hired but does not wish to
  work. The firm will not pay the worker if there
  is no work, but the firm cannot directly observe
  the workers effort level or output.
• Players The worker, the firm
• Strategy Work or not work, monitor or not
  monitor
• Payoffs Work pays 100, but the workers
  reservation wage is 40
• Worker can produce 200 in revenue, but it costs
  80 to monitor

24
The Shirking Game, Cont.

• There is no dominant strategy, or iterated
  dominant strategy.
• There is also no clear Nash Equilibrium. In
  other words, no combination of actions makes both
  sides happy given what the other side has chosen.
• There are many mixed strategies. The worker could
  work with probability (p) of 0.7, 0.6. 0.25,
  etc... The same is true for the firm. Which
  mixed strategy should they choose?
• If the worker is most likely to shirk, the firm
  should monitor. Likewise, if the firm is more
  likely to monitor, the worker should work. In
  any scenario, no Nash equilibrium will be found.
  The key is to find a strategy that makes the
  opponent indifferent to his/her potential
  choices.
• A person is indifferent when the expected return
  from action A equals the expected return form
  action B.

25
Solving the Shirking Game

• How much should the firm monitor?
• E(work) 60p 60(1-p) 60
• E(shirk) 0p 100(1-p) 100 - 100p
• 100 - 100p 60
• 40 100p
• p .40
• The worker is indifferent when the probability of
  monitoring is 40 and the probability of not
  monitoring is 60.
• How much should the worker work?
• E(monitor) 20p -80(1-p) 100p - 80
• E(Not monitor) 100p -100(1-p) 200p - 100
• 100p -80 200p - 100
• 20 100p
• p .2
• The firm is indifferent when the probability of
  working is 20 and the probability of not working
  is 80.
• How does the cost of monitoring and the workers
  reservation wage impact behavior?

26
Existence of Nash Equilibrium

• Every game with a finite number of players, each
  of whom has a finite number of pure strategies,
  possesses at least one Nash equilibrium, possibly
  in mixed strategies
• Final Note If the players have continuous
  strategies (as opposed to finite strategies) a
  pure strategy can be found with a reaction
  function.

27
The Football Game

• Scenario A game has come down to a final play.
  The 49ers are on the 2 yard line with 5 seconds
  to go. The current score is 20-16, with the
  Raiders in the lead. The 49ers have two choices,
  run or pass. The Raiders have two choices, defend
  against the run or defend against the passes.
• Players 49ers, Raiders
• Strategy Play Pass or Run, Defend Pass or Run
• Payoffs Probability of success given choices

28
The Football Game, cont.

• There is no dominant strategy, or iterated
  dominant strategy.
• There is also no clear Nash Equilibrium. In
  other words, no combination of actions makes both
  sides happy given what the other side has chosen.
• Hence this is a mixed strategy game.
• Remember, a person is indifferent when the
  expected return from action A equals the expected
  return form action B.

29
Solving the Football Game

• Should the 49ers run or pass?
• E(D run) 70p 20(1-p) 2050p
• E(D pass) 30p 80(1-p) 8050p
• 20 50p 80 50p
• 100p 60
• p .60
• The Raiders are indifferent when the 49ers run
  60 and pass 40 of the time.
• Should the Raiders defend the run or pass?
• E(run) 30p 70(1-p) 70 40p
• E(pass) 80p 20(1-p) 60p 20
• 70 40p 60p 20
• 50 100p
• p .5
• The 49ers are indifferent when the Raiders defend
  the run 50 of the time.

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Who will win the game?

• The probability that the 49ers will win the game
  the Nash Equilibrium strategies are adopted
  equals
• 0.6 0.5 30 0.6 0.5 70 0.4 0.5 80
  0.4 0.5 20 50
• The 49ers have a 50 chance of winning this game
  when each team adopts their equilibrium
  strategies.

31
The Football Game, new payoffs.

• How does changing the expected payoffs alter the
  probabilities that each team will take each
  action?
• The 49ers have a very good chance of scoring if
  they pass, and the Raiders play run defense.
• Outcome of the game
• 49ers will run with a probability of 4/7
• Raiders will play the run with a probability of
  2/7

32
Who will win the game now?

• The probability that the 49ers will win the game
  the Nash Equilibrium strategies are adopted
  equals
• 4/7 2/7 40 3/7 2/7 90 4/7 5/7 70
  3/7 5/7 50 61.4
• The 49ers have a 61.4 chance of winning this
  game when each team adopts their equilibrium
  strategies.

33
The Voting Game

• Non-intuitive game theory voting paradoxes
• Scenario Three economist need to decide how much
  math to require for economics majors. The
  options are
• 1) require no math
• 2) require one semester calculus
• 3) require two semesters calculus
• Preferences of each professor Dr. Vaitheswaran
  (V) LgtMgtH
• Dr. Berri (B) MgtHgtL
• Dr. Wu (W) HgtLgtM
• V is the chair of the committee, and V has the
  power to break any ties. Voting will be done
  simultaneously by secret ballot.
• Naive voting Professors ignore that it is a game
  and simply vote their preferences.
• Outcome V breaks the tie as the chair and the
  students at Coe have no math requirement.

34
The Voting Game, Cont.

• On the left are the outcome of the game, given
  each possible combination of votes for B and W,
  and each vote for V.
• The outcome in bold is the preferred outcome for
  V.
• V has a weakly dominant strategy (L). In three
  instances, Vs vote would be irrelevant,
  therefore V would not have a preference. In
  every other instance, V would maximize his
  utility by voting (L). From this we can
  conclude that V will vote (L).

35
The Voting Game, Cont.

• Voting for (L) is weakly dominated by (H) and
  (M), since this is the least of Bs preferences.
• Therefore, B will not choose (L), and we can
  eliminate this option.

36
The Voting Game, Cont.

• For W, (M) is weakly dominated by (H) and (L).
  Given this, W will choose (H) in every instance,
  so (H) is weakly dominant.
• The outcome of the game then is as follows
• V will vote L
• W will vote H
• B will vote H
• The students at Coe will thus have a high math
  requirement, exactly the opposite
• of what the chair wants.

37
The Good, The Bad, and the Ugly

• Scenario Three gunfighters in a gun fight. The
  winner gets the gold.
• Players Good is the fastest, Bad is the second
  fastest, and Ugly is the slowest at firing a gun.
• Each gunfighter only gets one shot, if he is not
  killed by a faster person. The winner gets the
  gold. If two people survive, the two agree to
  split the gold.
• All three gunfighters know the skill level of
  their opponents.
• Potential Actions Shoot at one of the remaining
  players.

38
The Good, The Bad, and the Ugly, cont.

• Ugly has a dominant strategy. If Ugly aims at
  Good, he is always better off than when he aims
  at Bad.
• Bad has the same dominant strategy. Aiming at
  Good results in a higher payoff than aiming at
  Ugly.
• Hence, in this game, the fastest gunfighter is
  killed.