Decisions with conflicts of interest: Games and strategy

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Decisions with conflicts of interest Games and
strategy

• Adding a decision maker with imperfectly aligned
  interests.

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A game has

• A finite list of players
• Each player has a strategy set from which she can
  choose actions
• The outcome of the game is defined by the
  strategy profile that consists of all strategies
  chosen by the players.
• Players have preferences over the outcomes of the
  play. What matters to players are outcomes, not
  actions.
• Both players are rational, and both know the
  other is rational

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Interactive Decision Theory

• Decision theory
• You are self-interested and selfish
• Game theory
• So is everyone else
• If its true that we are here to help others,
• then what exactly are the others here for?
• - George Carlin

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Some vocabulary

• Zero sum
• Non-zero sum
• Strategy
• Pure strategy
• Mixed strategy
• Equilibrium
• Solution
• Dominated Strategies

• Game trees
• Extensive form
• Normal form
• Strategic form
• Nash equilibrium
• Non-cooperative
• Cooperative
• Perfect/imperfect information
• Complete/incomplete information

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Rules of the Game

• The strategic environment
• Players
• Strategies
• Payoffs
• The rules
• Timing of moves
• Nature of conflict and interaction
• Informational conditions
• Enforceability of agreements or contracts
• The assumptions
• Rationality
• Common knowledge

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The Assumptions

• Rationality
• Players aim to maximize their payoffs
• Players are perfect calculators
• Common knowledge
• Each player knows the rules of the game
• Each player knows that each player knows the
  rules
• Each player knows that each player knows that
• each player knows the rules
• Each player knows that each player knows that
  each player knows that
• each player knows the rules
• Each player knows that each player knows that
  each player knows that each player knows that
  each player knows the rules
• Etc. etc. etc.

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Two person, zero-sum games

• Describe situations in which two individuals are
  in complete opposition to each other, where ones
  gain is the others loss.

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Two-person, strictly competitive games

• Player 2

(1, -1)
(-3, 3)
1
-3
Player 1
(-4, 4)
-4
(0, 0)
0
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Two person, zero-sum

• Play is simultaneous
• Perfect information
• Strict competition
• Players know one anothers possible strategies,
  outcomes and preferences
• No communication
• Everyone knows all of the above and is rational.

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Is there a solution?

• Every zero-sum game has a value that each player
  can guarantee him or herself (-1.5).
• All finite games have at least one Nash
  equilibrium.
• A Nash equilibrium captures the idea of
  equilibrium. Both players know what strategy the
  other player is going to choose, and no player
  has an incentive to deviate from equilibrium play
  because her strategy is a best response to her
  belief about the other players strategy.

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Equilibrium

• What is likely to happen
• when rational players interact in a game?
• Type of equilibrium depends on the game
• Simultaneous or sequential
• Perfect or limited information
• Concept always the same
• Each player is playing the best response to other
  players actions
• No unilateral motive to change
• Self-enforcing

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Prisoners Dilemma
Player 2
3, 3
-1, 4
Player 1
4, -1
0, 0
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Equilibrium

• Nash Equilibrium
• A set of strategies, one for each player, such
  that each players strategy is best for her given
  that all other players are playing their
  equilibrium strategies
• Best Response
• The best strategy I can play given the strategy
  choices of all other players
• Everybody is playing a best response
• No incentive to unilaterally change my strategy

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Prisoners Dilemma

• Each player has a dominant strategy
• Equilibrium that arises from using dominant
  strategies is worse for every player than the
  outcome that would arise if every player used her
  dominated strategy instead
• Private rationality ? collective irrationality
• Goal
• To sustain mutually beneficial cooperative
  outcome overcoming incentives to cheat

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Stackelberg Competition
Games of Chicken
Player 2
Enter
Dont enter
Fight
-1, -1
2, 0
Player 1
Accommodate
1, 1
2, 0
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Importance of Order

• Two equilibria exist
• ( Enter, Accommodate )
• ( Dont enter, Fight )
• Only one makes temporal sense
• Fight is a threat, but not credible
• Not sequentially rational
• Simultaneous outcomes may not make sense for
  sequential games.

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Game Theory

• Life must be understood backward, but it must
  be lived forward.
• - Soren Kierkegaard
• Mike Shor
• Lecture 4

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Sequential Games
The Extensive Form
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Looking Forward

• Entrant makes the first move
• Must consider how monopolist will respond
• If enter
• Monopolist accommodates

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And Reasoning Back

• Now consider entrants move
• Only ( In, Accommodate ) is sequentially rational

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Solving Sequential Games

• Start with the last move in the game
• Determine what that player will do
• Trim the tree
• Eliminate the dominated strategies
• This results in a simpler game
• Repeat the procedure

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Up and Down
3, 2
1, 3
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Identifying the Equilibrium

• Pure strategy equilibrium
• Consider mixed later
• Dominance
• Dominance solvable
• Only one dominant strategy
• Successive elimination of dominated strategies
• Cell-by-cell inspection

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Normal (Strategic) Form
PLAYERS
STRATEGIES
PAYOFFS
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A Strategic Situation

• Two firms competing over sales
• Time and The Economist must decide upon the cover
  story to run some week.
• The big stories of the week are
• A presidential scandal (labeled S), and
• A proposal to deploy US forces to Grenada (G)
• Neither knows which story the other magazine will
  choose to run

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One Dominant Strategy

• Who has a dominant strategy?
• Assume it will be played!
• Other player can plan accordingly.

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Dominated Strategies

• For The Economist
  
• G dominant S dominated
• Dominated Strategy
• There exists another strategy which always does
  better regardless of opponents actions

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Successive Elimination of Dominated Strategies

• If a strategy is dominated, eliminate it
• The size and complexity of the game is reduced
• Eliminate any dominant strategies from the
  reduced game
• Continue doing so successively

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Example Tourists Natives

• Two bars (bar 1, bar 2) compete
• Can charge price of 2, 4, or 5
• 6000 tourists pick a bar randomly
• 4000 natives select the lowest price bar
• Example 1 Both charge 2
• each gets 5,000 customers
• Example 2 Bar 1 charges 4,
• Bar 2 charges 5
• Bar 1 gets 300040007,000 customers
• Bar 2 gets 3000 customers

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Tourists Natives

• 
  in thousands of dollars

Bar 2
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Successive Elimination of Dominated Strategies

• Does any player have a dominant strategy?
• Does any player have a dominated strategy?
• Eliminate the dominated strategies
• Reduce the normal-form game
• Iterate the above procedure
• What is the equilibrium?

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Successive Elimination of Dominated Strategies
Bar 2
5
4
2
2
14
,
15
14
,
12
10
,
10
,
,
,
4
Bar 1
Bar 1
28
,
15
20
,
20
12
,
14
,
,
,
5
25
,
25
15
,
28
15
,
14
,
,
,
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No Dominated Strategies

• Often there are no dominated strategies
• Or reducing the game is not sufficient
• There may be multiple equilibria
• Method
• Cell-by-cell inspection
• Ask
• Is each player playing the best response to the
  other player?

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Zero-Sum Game
Column 3
Column 1
Column 2
-3
0
4
Row 1
3
1
2
Row 2
Row 3
-1
5
-2
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Next time . . .

• Voting behavior
• Mixed strategies (keep them guessing)
• Cooperation without verbal communication?

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Game Theory

• I Used to Think I Was Indecisive
• - But Now Im Not So Sure
• Anonymous
• Mike Shor Lecture 5