These notes were prepared based on the lectures and supplemented by information from textbooks and tutorials where parts of the lecture were unclear. Graphs, equations, and bullet-point explanations included. Prepared by a first class Economics and Management student for the FHS Microeconomics pape...
MT4 Game Theory
Outline
Lecture 9: Definitions, dominance, Nash equilibrium
Lecture 10: Dynamic games in extensive form and subgame-perfect equilibrium (SPE)
Lecture 11: Finite and infinite repetition, Folk Theorem, Nash reversion
Goals
Define non-cooperative games in strategic form
Dominance-based solution concepts
o strictly dominant strategies
o iterated elimination of strictly dominated strategies
Nash equilibrium: existence, finding them all
Dynamic games and subgame perfection
Repeated games and “folk theorems”
Lecture 9: Strategic form games
Introduction
What is game theory
Game theory studies multi-agent optimization problems
o multiple agents take actions that affect the welfare of all
o contrast with methodology in consumer & producer theory: constrained optimization,
non-strategic price-taking behaviour
Various solution concepts (non-cooperative equilibria, cooperative solutions/ values) are used
for positive and normative purposes
o positive: what will happen, normative: what should happen
Applications:
o markets with few, strategic firms choosing prices, quantities, ads
o competition, cooperation, compromise among political entities
o bargaining, cost or reward sharing, matching, auctions
o communication (expert to policy makers, central bank to all)
History
Cournot (1838) described a model of oligopolistic competition as a “quantity-setting game”
Neuman (1928) defined strategic games, mixed strategies, proved existence of minmax
equilibrium in zero-sum game
Nash (1950) defined the Nash equilibrium
Selten (1965) introduced the concept of perfection in dynamic games
Harsanyi (1967) introduced bayesian equilibrium under incomplete info
A motivating example
A ban on cigarette TV ads, industry profits increased
Explanation: TV ads never actually increased total industry sales. Tobacco companies were just
“stealing” consumers from each other. The ban helped them get out of this inefficient situation
, Two game-theoretic models of the example
o Model 0: Commitment in the Prisoner’s Dilemma
without ads each firm gets $50m in profits, but each can spend $20m on
advertising to steal $30m in profits from the other
optimal for each to play Ads, ban allows commitment to No Ads
o Model 1: Shi Qi (2013)
Heterogeneous brands accumulate goodwill via ads; TV ban forces switch to
print ads whose efficacy they learn over time
“Sophisticated econometrics” used to estimate effectiveness of ads, brand
depreciation, effect of ban on market structure
What's a game?
A game in strategic form is given by
o the set of players
o for each player a set of possible strategies
o and for each player a payoff in every outcome
Each player picks a strategy simultaneously; the strategy profile (=one strategy from each
player) determines the outcome; agents’ preference orderings over the outcomes are
represented by payoffs
Each agent is fully aware of the game (players, strategies, payoffs), acts to maximize own payoff
knowing others do the same
This is the simplest model of a game, other approaches are known too
Caveats for the modeller
A strategic situation is often described verbally; the first hurdle is to write down a model faithful
to the agents’ perception of the situation
Example 1: Donation Game/ Prisoner's Dilemma
o “Ann and Bob participate in an experiment. Each gets a £30 budget. Then each
simultaneously decides whether s/he gives up £10 to let the other have £30 more.”
o
o Are we sure each only cares about his or her own take-home cash? (Altruism, inequality
aversion, maybe secret side-contracts?).
o This could affect the payoffs in the payoff matrix above
Framing (the story that motivates the game written down as “players, strategies, payoffs”) will
not play any role in our analysis of the game (eg. Prisoner's dilemma can be framed as the
donation game), but it may affect how real-life individuals behave in a given strategic situation
o In applied work it is an important job for the modeller to capture all relevant aspects of
the strategic situation in the abstract game, the way it is likely perceived by the actual
players (adjust the payoffs to fit perception changes arising from framing)
, Externalities (e.g., envying other players in a given outcome, or feeling empathy), concerns
about counterfactuals (outcomes that could have occurred but did not), and so on, should all be
incorporated in the payoffs.
Dominance-based solution concepts
A strictly dominant strategy is one that yields a greater payoff to the player than any other
strategy no matter what the other players do
The Prisoner’s Dilemma features a strictly dominant strategy for each player, namely “Not” (or
Ads), because (60, 30) > (50, 20)
Playing a strictly dominant strategy is extremely compelling, requires “no game theory” (i.e.,
reasoning about what others might do)
In games where each agent has a dominant strategy our prediction is that they will play it
o If behaviour in an experiment contradicts this, then probably the model does not reflect
how agents understand the experiment
Iterated strict dominance
Example 2: modification of Example 1
o
o Not a Prisoner’s Dilemma
o No dominant strategy for Ann (row), but R strictly dominates L for Bob, so he will only
ever play R. Knowing this, Ann plays B
o Explanation: even if a player does not have a dominant strategy (Ann), if the other
player has a dominating strategy (Bob plays R), then we can know Ann's strategy by
looking at the unique best response to Bob's strictly dominant strategy
Iterated Elimination of Strictly Dominated Strategies (IESDS) or Iterated Strict Dominance (ISD):
o Eliminate strictly dominated strategies (if any) for each player
o This may “unlock” further rounds of elimination
o Repeat until no strategy is strictly dominated in reduced game
Simplified Cournot duopoly model: Discrete Cournot
o 2 firms simultaneously set quantities q1 and q2
o The good’s price is p = 9 - (q1 + q2)
o No production cost, so firm i's profit: pqi = (9 - q1 - q2)qi.
o Only three possible production levels: H=4, M=3, L=1
o
Payoff matrix based on profits
Row’s payoff written first
No dominant strategy but we have a dominated strategy (L)
For the column player, H dominates M if the other player plays L but M
dominates H if the other player plays M or H.
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