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Game Theory. Section 1. 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. What Defines Game Theory?.

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game theory
Game Theory

Section 1

agenda
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
what defines game theory
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
exercise moving to equilibrium
Exercise: Moving to Equilibrium

Player 2

A Game in Normal Form

Avg payoff for playing with prob distribution p, given others play strategies with prob distribution q

Expected Value:

three or four important concepts
Dominant Strategy

One best strategy no matter what your opponent does

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

Three or Four Important Concepts

Beliefs: Probability distribution over strategies of other players

Nash equilibrium: When each player is playing a best response to the strategies of the others

key terms and definitions
Key Terms and Definitions
  • Normal form and extensive form
  • Nodes (initial, terminal)
  • Information sets
  • Strategy: complete contingent plan
    • One for every information set
  • Equilibrium: a stability concept
  • Welfare
  • Pareto optimality: no free lunch
formal theory what s it good for
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.

The world of our constructs is therefore the desired island that is exempt from the flux of blind and aimless causation.

applying formal theory
Applying Formal Theory
  • Goal (research question)
  • Story (model)
  • Formalization (abstraction)
  • Analysis (manipulation)
  • Solution (definition of equilibrium)
  • Translation (external and internal validity)

Game Theory is just one of several formal modeling approaches

General Equilibrium: interactions among many (infinite) agents, where any one agent’s actions have no effect on other agents.

example alliance formation
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 positively associated with probability of winning
    • Each is rational and has its own interests at heart

Example taken from Altfield and Bueno de Mesquita 1979

formalization
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
    • 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 neutrality is zero
  • B vs. nature: B moves, then either A or C wins
analysis whither alliances
Analysis: Whither Alliances?
  • Adopt the perspective of a player: B
  • B’s utility from an alliance with A = PBA(UBA) + (1-PBA)(UBC) - KBA
  • B’s utility from an alliance with C = PBC(UBC) + (1-PBC)(UBA) - KBC

(1)

(2)

  • What if equation 1 > equation 2?
  • What if equation 2 > equation 1?
solution equilibrium
Solution: Equilibrium

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...

[b/(a + b + c)] (UBA -UBC) = (KBA -KBC)

translation validity
Translation: Validity
  • What is the point at which B is indifferent?

[b/(a + b + c)] (UBA -UBC) = (KBA -KBC)

  • [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

  • Calibration, 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
  • [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 then B would rather partner with A
  • What real-world events might prompt this?