Strategic Form Games

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Strategic Form Games

• Game Theory, Winter 2007
• Guilherme Carmona

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Focus of GT Interdependence

• What will each individual guess about the others
  choices?
• What action will each person take?
• What is the outcome of these actions?
• Does it change if the group interacts more than
  once?
• Does it change if each person is unsure about the
  characteristics of the others?

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GT is a formal way to analyze interactions among
a group of rational agents who behave
strategically
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• Group there are more than one DM
• Interaction the choice of each player affects
  directly other(s) player(s)
• Strategic each player accounts for this
  interdependence when choosing
• Rational each player chooses her best action

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Strategic Form Game

• Game in normal form is G
  (N,S1,,Sn, u1,,un)
• Players N1,,n
• Player is strategy sets Si
• Player is payoffs ui(s1,,sn) ui(s)

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Notation

• A particular strategy for player i si
• A strategy choice s(s1,...,sn)(si,s-i)
• Examples

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Notions of Dominance

• Usefulness of strategies
• Strongly / weakly dominant
• Strongly / weakly dominated
• Undominated
• Related equilibrium concepts
• Dominant strategy solution
• Iterated Elimination of Dominated Strategies or
  Dominance Solvability

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Dominant strategies

• A strategy si is strongly dominant ifu(si,s-i)
  gt u(si,s-i) ? si ? si, si? Si, s-i? S-i
• A strategy si is weakly dominant ifu(si,s-i)
  u(si,s-i) ? si? Si, s-i? S-i,and for each si ?
  si, si? Si there is somes-i ? S-i such that
  u(si,s-i) gt u(si,s-i)gt si weakly dominates
  every other strategy si

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Dominant Strategies (2)

• Weakly dominant strategies are rare
• Strongly dominant strategies are very rare
• For each player there is at most one!
• If every player has a dominant strategy, then the
  game has a dominant strategy solution
  everybody plays their dominant strategy!
• Advantage very nice if exists
• Disadvantage almost never exists

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Dominated strategies

• A strategy si? Si is strongly dominated if there
  is a strategy si? Si such thatu(si,s-i) lt
  u(si,s-i) ? s-i? S-i
• A strategy si?Si is weakly dominated if there is
  a strategy si? Si such thatu(si,s-i) ?
  u(si,s-i) ? s-i? S-i,and there is some s-i ?
  S-i such that u(si,s-i) lt u(si,s-i)
• Rational players will not play dominated
  strategies?

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Dominated Strategies (2)

• Rational players will definitely not play
  strongly dominated strategies
• Rational players may play weakly dominated
  strategies, even in equilibrium

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Nash Equilibrium

• Central concept of game theory
• Proposes strategy bundle s
• No player deviates given s-i
• Likely to stayequilibrium

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A formal definition

• A strategy combination s (s1,,sn) ? S is a
  Nash equilibrium if, for each player i,
  ui(si,s-i) ui(si,s-i) ? si ? Si
• Weak inequality!
• Indifference is enough
• Weakly dominated strategies can take part in a
  Nash equilibrium
• Notion with gt strict NE

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Interpretation

• Stable strategy combination, no deviations if
  other player expect s-i
• Either through thinking
• Or by proposal
• Or convergence through trial and error
• Problems
• How to find?
• Existence?
• Uniqueness or multiplicity?

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Relations betweensolution concepts

• Dominant strategy equilibrium
• DSE gt Nash Equilibrium
• Converse wrong
• IEDS
• NE gt survives IESDSConverse wrong
• NE may be eliminated by IEWDS

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Readings and exercises

• Readings ch. 3,4,5
• Exercises PS1 3, 9
• Exercises Dutta 5.1, 5.2, 5.3