2. Lecture 2: Static Games and Pure-
strategy Nash Equilibrium (PNE)
■ Review the definition of static games
■ Classic examples of static games
■ Best responses (definition and examples)
■ Dominance (strictly dominant, weakly dominant)
■ Pure-strategy Nash Equilibrium (PNE)
■ Zero-sum games
■ Iterated elimination of dominated strategies (IEDS) and its relation to PNE
■ Best-response dynamic
3. Review: Static games
■ In last lecture, we’ve seen the definition of static games
■ Some clarifications to better understand the definition…
■ i index each individual
■ a is a vector that represents all actions of players, with each ai represents the action of
individual i
■ Ak is the set of possible actions, with ak to be its realization
5. ■ Strategies versus Actions:
■ Action: any element from Ai for 𝑖 𝜖 𝑁
■ Strategy: a “plan of action". In static games a strategy is a particular element from Ai
, but notion of “strategy" is adaptable/ more general.
Review: Static games
10. Best responses
■ The best response captures the best choice of individual i when given other players
actions a-i
■ Intuition: the response returns higher utility than any other actions, and utility represents
the preference of the individual.
■ We use the notation “*” to denote the best response of each individual
12. Dominance: strict dominance
■ Intuition: no matter what the actions of the others, one strategy is always worse.
Therefore, in all situations, the individual will not choose the action. This is where the word
“dominance” comes from.
13. Dominance:
■ Consider the possibility to use dominant solvability as a concept to capture the
equilibrium we would like to research
■ What are the pros and cons if we would use the concept as solution?
16. Pure-Strategy Nash Equilibrium
■ Dominant solvability gives us a strong but rare situation, which possibly capture the
equilibriums in games
■ But it is so rare that would do little help in our analysis of the world
■ So we have a weaker version of equilibrium: pure-strategy Nash Equilibrium
■ The Nash equilibrium was named after American mathematician John Nash
■ In terms of game theory, if each player has chosen a strategy, and no player can
benefit by changing strategies while the other players keep theirs unchanged, then
the current set of strategy choices and their corresponding payoffs constitutes a
Nash equilibrium.
20. Zero-sum games
■ In game theory and economic theory, a zero-sum game is a representation of a
situation in which each participant's gain or loss of utility is exactly balanced by the
losses or gains of the utility of the other participants.
■ If the total gains of the participants are added up and the total losses are
subtracted, they will sum to zero.
■ Thus, cutting a cake, where taking a larger piece reduces the amount of cake
available for others, is a zero-sum game if all participants value each unit of cake
equally
21. Zero-sum games
■ Matching-pennies, or Chess/Checkers/Go and two-player card games as dynamic
games (later), give examples of zero-sum games.
■ Fact: Zero-sum games can have PNE, e.g., (T, L) in the game:
■ Many features of games not captured by zero-sum games (e.g., coordination
failures, public goods).
22. Iterated elimination of dominated strategies
(IEDS)
■ Iterated elimination of strictly-dominated strategies (IESDS):
■ Continue to eliminate strictly dominated strategies in this way, until you have a game
without any dominated strategies for any player.
23. Iterated elimination of dominated strategies
(IEDS)
■ The simple rule to do IESD: compare the values of strategies in each row/ column, if
each value of the other row/column is greater than the value in current row/column
respectively, then current row/column is dominated
24. Iterated elimination of dominated strategies
(IEDS)
■ Fact 1: For game Γ′ obtained from Γ via IESDS:
𝒂∗ ∈ 𝑁𝐸 Γ′ ⇔ 𝒂∗ ∈ 𝑁𝐸(Γ)
■ Iterated elimination of weakly-dominated strategies (IEWDS): as above but replace
“strict" with “weak".
■ Fact 2: For game Γ′ obtained from Γ via IEWDS:
𝒂∗ ∈ 𝑁𝐸 Γ′ ⇒ 𝒂∗ ∈ 𝑁𝐸(Γ)
■ However, the proposition (i.e. the converse of Fact 2) no longer holds
R is weakly dominated by L, but
not strictly dominated.
IEWDS leaves only (T,L) in the
end, while IESDS leaves all four
action states.
25. Best-response Dynamic
■ Intuition: the belief in current time period is derived from the actions from other players in
previous time period. The player takes action based on such belief.
■ Fact 3: Clearly, pure-strategy Nash equilibria give stationary points (“sinks") in multilateral
and unilateral best-response dynamics.
■ Fact 4: No guarantee all multilateral/unilateral best-response dynamics converge, even
when a pure-strategy Nash equilibrium exists...
27. Summary
■ Definition of static games
■ Best responses
■ Dominance (strictly dominant, weakly dominant)
■ Pure-strategy Nash Equilibrium (PNE)
■ Zero-sum games
■ Iterated elimination of dominated strategies (IEDS) and its relation to PNE
■ Best-response dynamic
