• Weakly dominant versus strongly (strictly dominant) strategies
Mixed Strategy
• Moves are chosen randomly from the set of pure strategies
• Every simultaneous move game has a Nash equilibrium in mixed strategies
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Formalization
• States are identified as (A, B, C)
• Their capabilities are (a,b,c)
– If B helps A, probability that A wins is (a + b)/(a + b + c) = PBA
Three or Four Important Concepts
Beliefs: Probability distribution over strategies of other players
Dominant Strategy
• One best strategy no matter what your opponent does
The world of our constructs is therefore the
desired island that is exempt from the flux
of blind and aimless ch ausation.
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Applying Formal Theory
• Goal (research question) • Story (model) • Formalization (abstraction) • Analysis (manipulation) • Solution (definition of equilibrium) • Translation (external and internal validity)
GOV 2005: Game Theory
Section 1
Alexis Diamond adiamond@
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Agenda
• What is game theory? • Exercise: convergence to equilibrium • Key terms and definitions • Formal theory: what Leo has to say about it • How do you do it (using formal theory) • Application: Alliance Formation • Final thoughts
– If B helps C, probability that C wins is (b + c)/(a + b + c) = PBC
• Amount that B values a victory by A is UBA • Cost to B of helping A is KBA
– Cost of neutibration, comparative statics, case studies
are all ways of assessing model validity
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Final thoughts
• Given B is indifferent to allying when [b/(a + b + c)] (UBA -UBC) = (KBA -KBC), and
What is the point at which B is indifferent? PBA(UBA) + (1-PBA)(UBC) - KBA ) = PBC(UBC) + (1-PBC)(UBA) - KBC )
Simplifies to...
(PBA + PBC -1)(UBA -UBC) = (KBA -KBC) Simplifies to...
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What Defines Game Theory?
Formal Analysis of Strategic Settings
• Mathematically precise • List of players • Complete description of allowable moves • Description of information • Specification of how actions cause outcomes • Specification of players’ preferences
positively associated with probability of winning – Each is rational and has its own interests at heart
Example taken from Altfieldhand Bueno de Mesquita 1979
Game Theory is just one of several formal modeling approaches
General Equilibrium: interactions among many (infinite) agents, where any one agent’s actiohns have no effect on other agents.8
[b/(a + b + c)] (UBA -Uh BC) = (KBA -KBC) 12
Translation: Validity
• What is the point at which B is indifferent? [b/(a + b + c)] (UBA -UBC) = (KBA -KBC)
Nash equilibrium: When each player is playing a best response to
the strategiehs of the others
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Key Terms and Definitions
• Normal form and extensive form
• B vs. nature: B moves, then either A or C wins
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Analysis: Whither Alliances?
• Adopt the perspective of a player: B
• B’s utility from an alliance with A = PBA(UBA) + (1-PBA)(UBC) - KBA (1)
• [b/(a + b + c)] = resources B can contribute • (UBA -UBC) = B’s motivation for A vs. C • (KBA -KBC) = B’s costs for A vs. C
Equilibrium, perturbed: • If the left-hand side is > the right-hand side
• Pareto optimality: nohfree lunch
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Formal Theory: What’s it good for?
“We understand only what we make” --Leo Strauss
We have absolutely certain or scientific knowledge only of those subjects of which we are the causes… The construction must be conscious construction; it is impossible to know a scientific truth without knowing at the same time that we have made it.
Example: Alliance Formation
• Goal:
– Gain insight into dynamics of power and motivation – What motivates countries to form alliances?
• Story:
– Setting: 3 countries fighting a war – Each has power (military capability), which is
then B would rather partner with A • What real-world evehnts might prompt this?14
• [b/(a + b + c)] = resources B can contribute • (UBA -UBC) = B’s motivation for A vs. C • (KBA -KBC) = B’s costs for A vs. C
The decision to help in a dispute depends on one’s ability to influence the outcome, one’s level of motivation, and the costliness of getting involved
Player 1 Up
Player 2
Left -8, 10
Right -10, 9
Down
7, 6
6, 5