Solving Games: Nash Equilibrium

1
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.

2
Backward Induction

• 1. Start at terminal nodes of the game, and trace
  back to its parent.
• 2. Find optimal decision for that player at that
  decision node by comparing payoffs the player
  receives from each terminal node. Record this
  choice, it is part of the players optimal
  strategy.
• 3. Prune from the game tree all the branches that
  originated from 1. Attach to each of these new
  terminal nodes the payoffs received when the
  optimal action is taken at this node.
• 4. A new game tree exists and is smaller than the
  original, if there are no decision nodes, you
  are done.
• 5. If there are decision nodes, apply 1 thru 4
  until no decision nodes are left.
• 6. For each player collect the optimal decisions
  at every node. This collection constitutes that
  player's optimal strategy in the game.

3
Backwards Induction, cont.

• Review the software game in 5.1 Use backwards
  induction to determine both players optimal
  decisions.
• Is this a Nash Equilibrium? Yes.
• Why? All backwards induction solutions are Nash
  equilibrium.

4
Threats Credible vs. Incredible

• Inability to make enforceable commitment, not no
  communication drive the above results.
• Suppose Microcorp tells Macrosoft it will always
  clone the software no matter what advertising
  campaign Macrosoft employs. This is a Threat.
• A threat occurs when one player attempts to get
  other players to believe it will employ a
  specific strategy.
• Given this threat, Macrosoft would be wise to
  ignore this threat, because it is not credible.
• A threat by a player is not credible unless it is
  in the player's own interest to carry out the
  threat when given the option. Threats that are
  not credible are ignored.

5
Threats, cont.

• A binding threat is one in which a player can not
  back down.
• FIGURE 5.2 gives a binding threat game tree.
  Note Idle is dropped for simplicity.
• What is Macrosofts optimal strategy?MA only if
  Microcorp threatens EMA, EWOM and WOM for
  all other strategies
• Note There is a relationship between credible
  threats and Nash equilibrium. If Macrosoft
  believes that Microcorp will employ ENTERMA,
  ENTERWOM, then Macrosoft should play WOM. It is
  also true that if Microcorp believes that
  Macrosoft plays WOM it should play ENTERMA,
  ENTERWOM, which is a Nash Equilibrium. Yet it
  is a Nash equilibrium that relies on an
  incredible threat.

6
Subgames

• A subgame is essentially a smaller game within a
  larger game with two special properties.
• Once the players begin playing a subgame, they
  continue playing the subgame for the rest of the
  game.
• The players all know when they are playing the
  subgame.
• A subgame has one initial node, the subroot.

7
Subgames, cont.

• The subgame Gs of the game GT is a game
  constructed as follows
• Gs has the same players as GT, although some of
  these players may not make any moves in Gs
• The initial node of Gs is a subroot of GT, and
  the game tree of Gs consists of this subroot, all
  its successor nodes, and the branches between
  them.
• The information sets of Gs consist of those
  information sets of GT that contain a decision
  node of Gs.
• The payoffs of each player at the terminal nodes
  of Gs are identical to the payoffs in GT at the
  same terminal nodes.
• These four conditions say that the set of
  players, order of play, set of possible actions
  and the information sets in the original game are
  preserved in the subgame.

8
Subgames, cont.

• This definition implies that every game is a
  subgame of itself.
• FIGURE 5.3 How many subgames exist? Four.
• Playing any subgame is common knowledge to the
  players. Thus rational behavior for the players
  in the overall game should also appear rational
  when viewed from the perspective of this subgame.
• WOM, ENTERMA, ENTERWOM, STAY OUTIDLE is
  a Nash equilibrium that involves an incredible
  threat by Microcorp to enter no matter what
  advertising strategy Macrosoft uses.
• The subgame in Figure 5.3 at D2 has only one Nash
  equilibrium which Microcorp chooses Stay Out.
• Yet the strategy profile selects Microcorp to
  enter.
• It is this non-optimality in the subgame which
  makes the threat incredible.

9
Subgame-Perfect Equilibrium

• A Nash equilibrium strategy that remains a Nash
  equilibrium when applied to any subgame is
  subgame-perfect.
• A strategy profile is a subgame-perfect
  equilibrium of a game G if this strategy profile
  is also a Nash equilibrium for every subgame of
  G.
• In games of perfect information, subgame-perfect
  equilibria are those found via backwards
  induction.
• Backwards induction eliminates all the incredible
  threats, and all actions are a Nash equilibrium,
  and subgame-perfect equilibrium.
• REVIEW THE SOFTWARE GAME WITH DIFFERENT PAYOFFS.

10
Games with Perfect Information

• Games with Uncertain Outcomes involve a new
  player, Nature. When we come across an example
  of a game with Nature, we will return to the
  details of this type of game.
• Games with Perfect Information and Continuous
  Strategies
• How much output should I produce?
• How much advertising should I purchase?
• The Stackleberg Duopoly

11
Chapter Eleven Nash Equilibrium II

• Modeling Simultaneous-Move Games
• Review Figure 11.1
• The model appears sequential.
• Since the decision nodes for Rusty are in the
  same information set, the model is simultaneous.
  
• Remember, simultaneous is not a reference to
  chronological time, but rather the fact that each
  player does not know the other players move
  before she/he makes her/his move.
• Hence, Figure 11.2 is equivalent to Figure 11.1

12
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. Review Table 11.4
• 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.

13
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.

14
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)
  

15
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.

16
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

17
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.

18
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.

19
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.

20
Example of Iterated Dominance

• Down is Firm 1, Across is Firm 2

21
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.

22
A Lack of Dominance

• Down is Player 1, Across is Player 2

23
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.

24
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

25
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

26
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.

27
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?

28
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.

29
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

30
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.

31
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.

32
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.

33
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

34
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.

35
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) LMH
• Dr. Berri (B) MHL
• Dr. Wu (W) HLM
• 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.

36
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).

37
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.

38
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.

39
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.

40
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.