ECOS3012 Lecture 2
ECOS3012 Lecture 2
August 17, 2021
Recap
Goal of game theory:
communicate a strategic situation (define a game)
make predictions (solve a game)
Solution concepts
Iterated elimination of strictly dominated strategies (IESDS)
Nash equilibrium (NE)
Recap: key concepts
Strictly dominated strategy ( is SDed by )
No matter what other players’ strategies are, generates a strictly lower payoff than
Best response
Given other players’ strategies, the strategy that generates the highest payoff
Nash equilibrium
All players best respond
Recap: IESDS vs NE
Why must an NE survive IESDS?
Let the NE be .
Suppose it is deleted in a step when we delete , which means is not yet deleted
We cannot have a strategy such that for all .
Because we know is a best response to .
This week: four applications of NE
Cournot competition
Bertrand competition
Hotelling competition
Tragedy of the commons
Goals:
Describe a strategic situation using a game
Work out best response functions (correspondences)
Find out Nash equilibrium
Cournot competition
Game:
Players: N = {1, 2}
Strategy sets: ,
Payoff functions: Profit = Revenue – Cost
Best responses
Strategic situation:
Two firms producing homogeneous good
Each chooses its own quantity: and .
Market determines the price (demand function):
Firms maximize their profit, marginal cost of production 10, no fixed cost
Cournot competition
Nash equilibrium:
Equilibrium profit:
Monopoly
If firms collude and behave as a monopoly (split the profit in half)
Comparison
Monopoly profit is higher
However: what is firm 2’s best response if firm 1 chooses a quantity of 22.5?
Bertrand competition
Strategic situation:
Two firms producing homogeneous good
Simultaneously chooses a price to charge:
Market purchases from the lower priced firm (evenly split if )
Demand function
Firms maximize profit, marginal cost = 10, no fixed cost
Game:
Set of players: N = {1, 2}
Strategy sets:
Payoff functions: Profit = Revenue – Cost
Bertrand competition
Unique Nash equilibrium:
Equilibrium profit: zero
Firms undercut each other by lowering their prices, until 0 profit
Cournot vs Bertrand
Cournot: industries where the quantity of output is difficult to adjust in the short-run (e.g. agriculture)
Firms compete in quantities
If firms are homogeneous, then each firm’s profit is lower than monopoly profit, but higher than 0
Increase quantity negative externality excess quantity
Bertrand: industries where quantity of output can be easily adjusted (e.g. restaurants)
Firms compete in price
If firms are homogeneous, profit = 0
In effect, duopoly = perfect competition
Hotelling model
Strategic situation:
Two vendors, each chooses a location to set up a shop on King Street
Customers are uniformly distributed along King Street, and goes to the closer vendor
Vendors maximize their market shares
Game:
Model King Street with a unit interval [0, 1].
Set of players: N = {1, 2}
Strategy sets:
A strategy denotes the location of vendor .
Payoff functions:
Hotelling model
Suppose vendor 1 chooses , what is vendor 2’s best response?
Suppose vendor 1 chooses , what is vendor 2’s best response?
Suppose vendor 1 chooses , what is vendor 2’s best response?
Nash equilibrium: .
Cournot vs Bertrand vs Hotelling
Cournot: competition in quantities
Bertrand: competition in price
Hotelling: competition in product attributes (e.g., location, size, style, functionality, sweetness)
The tragedy of the commons
Set of players: 20 farmers
Strategies: Each farmer chooses the number of goats to gaze on the village green
Payoff functions: for each farmer
Cost of owning a goat is 10
Total number of goats on the green:
For each goat: value of gazing on the green
The tragedy of the commons
Nash Equilibrium:
Social optimum:
The green is over-gazed.
Each farmer earns significantly lower than social optimum.
Why over-gazing:
Additional goat if
marginal benefit > (his own) marginal cost
Marginal benefit is enjoyed by him alone
Marginal cost is born by everyone
The tragedy of the commons
Similar examples: pollution, traffic congestion, shirk duties in a team project
Game theory: given the payoff specified in this example and similar examples, the only rational prediction is that people will overproduce/over-consume/under-perform.
A result of people being self-interested and rational.
How to fix it?
Change people’s payoffs. E.g., penalty/reward (monetary or moral)
/docProps/thumbnail.jpeg