Lecture 2: Static games with complete information II
1 Recap of last lecture
• Game theory is about making predictions in situations of strategic interaction
• We develop criteria to select the more likely outcomes out of the many possible out-
comes
• Criteria that we learned from last time
– Iterative elimination of strictly dominated strategies => often too weak
– Nash equilibrium: in a NE, each player chooses the best response to other player’s
equilibrium strategies, i.e., no player should have an incentive to choose a different
strategy
– NE is not perfect: it can eliminate too many or too few outcomes
– NE is not the unique solution concept
– Nevertheless, NE is, perhaps, the most reasonable selection criterion that economists
agree upon, and we will continue to focus on finding the Nash equilibria of static
games with complete information
2 Applications of Nash equilibrium
2.1 Cournot Model of Duopoly
2.1.1 Set up
• Players: two firms, 1 and 2
• Strategies: each firm chooses to product a homogenous product at quantity qi ≥ 0
• Payoffs: firms’ profits are determined in the following way
– Market demand has a downward slope: P (q1, q2) = 100− q1 − q2
– Total cost to product qi for each firm is Ci(qi) = 10qi for i = 1, 2
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– Profit for each firm is Revenue – Cost, i.e.
π1(q1, q2) = q1 (100− q1 − q2)− 10q1
π2(q1, q2) = q2 (100− q1 − q2)− 10q2
2.1.2 Nash equilibrium
• Recall: (q∗1, q∗2) is a Nash equilibrium if q∗1 is a best response to q∗2 and vice versa
• Solve
max
q1
π1(q1, q2) = q1 (100− q1 − q2)− 10q1
max
q2
π2(q1, q2) = q2 (100− q1 − q2)− 10q2
• Set π′i(qi) = 0 to get
(90− qj)− 2qi = 0
q∗1 =
1
2
(90− q∗2) , q
∗
2 =
1
2
(90− q∗1)
q∗1 = q
∗
2 = 30
π∗1 = π
∗
2 = 900
2.1.3 Monopolistic profit
• What is the highest feasible profit? If the firms collude and jointly behave as a
monopoly, they should choose the following strategy: let q1 = q2 = Q
max
Q
π(Q) = Q (100−Q)− 10Q
Qm = 45, qm1 = q
m
2 = 22.5
πm1 = π
m
2 = 1012.5 > 900
• Why is the better monopolist strategy not a Nash equilibrium? To show that an
allocation is not NE, we need to find profitable deviations. Suppose Firm 2 chooses
qm2 = 22.5, then firm 1 can strictly profit by deviating to a higher quantity. For
example,
π1(22.5, 22.5) = 1012.5 but π1(30, 22.5) = 1125
• See graph to illustrate convergence to the unique NE
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•
• The concept of NE is very similar to the concept of a steady state
2.2 Bertrand Model of Duopoly
• Cournot game is good to model an industry where the output and capacity are difficult
to adjust.
• When the capacity and output can be easily adjusted, firms can compete in price,
instead of quantities. => Bertrand game.
2.2.1 Set up
• Two players: firm 1 and firm 2
• Strategies: each firm chooses a price pi > 0
• Payoffs:
– demand for firm i is
∗ 100− pi when pi < pj
∗ 0 when pi > pj
∗ 100−pi
2
when pi = pj
– No fixed cost, marginal cost is 10
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2.2.2 Nash equilibrium
• Firm i’s best response to pj
– When pj > 10, πi(pi, pj) =
0 pi > pj
(pi − 10) · (100− pi) pi < pj 1 2 (pi − 10) · (100− pi) pi = pj – πi is maximized when pi = pj − � for some positive � close to zero. • Firms undercut each other by lowering price, until they both charge p = 10 (marginal revenue = marginal cost) and earn zero profit. • Unique NE: p1 = p2 = 10. In the NE, π1 = π2 = 0. 2.3 Compare Cournot vs. Bertrand • Bertrand: when homogeneous firms compete in price, duopoly = perfect competition • Cournot: when two homogenous firms compete in quantities, then each firm’s profit is below the monopoly profit, but above the zero-profit in perfect competition. [In the practice problem this week we will solve a Cournot game with infinitely many firms and show that the result converges to perfect competition.] • Another way to interpret firms’ low profits (lower than monopoly) in Cournot equilib- rium: each firm’s action generates externalities on the opponent firm. In the Cournot game, increasing quantity creates a negative externality for the competing firm. When there are negative externalities, agents overproduce. In essence this is not dramatically from other problems with negative externalities, such as pollution, traffic congestion. 2.4 Hotelling model of duopoly (Harold Hotelling, 1929) There are two vendors, i = 1, 2. Each vendor chooses a location to set up a shop on King Street. Suppose that we use the interval [0, 1] to model the entire King Street. Then, the strategy for each vendor is to choose a location xi ∈ [0, 1]. Suppose that the consumers are uniformly distributed along King Street, and go to whichever vendor is closer. A vendor’s profit is proportional to his market share, i.e., vendor i’s profit is πi(xi, xj) = 1 2 (xi + xj) xi < xj, 1 2 xi = xj, 1− 1 2 (xi + xj) xi > xj.
(a) Suppose vendor 1 chooses x1 = 0, what is vendor 2’s best response?
A: Choose x2 = �, for �→ 0.
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(b) Suppose vendor 1 chooses x1 = 12 , what is vendor 2’s best response?
A: Choose x2 = 12 .
(c) Find the Nash equilibria of this game.
A:
(
1
2
, 1
2
)
.
2.5 The tragedy of the commons
• First raised by William Forster Lloyd (1833), later cited by Garrett Hardin (1968)
2.5.1 Set up
• Players: 20 farmers
• Strategy: each farmer i chooses the number of goats gi to graze on the village green
• Payoff: for each farmer i
– cost of owning a goat is 10
– Let the total number of goats grazed on the green be G = g1 + g2 + … + g20.
For each farmer’s each goat, the value of grazing that goat on the green is
v(G) = 10000−G2
–
2.5.2 Nash equilibrium
• Each farmer faces the same problem. It’s sufficient to solve farmer 1’s best strategy
max
g1
g1 · [10000− (g1 + g2 + …+ g20)2]− 10g1
[10000− (g∗1 + g2 + …+ g20)
2]− 2g∗1(g
∗
1 + g2 + …+ g20)− 10 = 0
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• In equilibrium, since every farmer faces the identical problem, we must have g∗1 = g∗2 =
… = g∗20 = g
∗. Therefore, the NE must satisfy
[10000− (20 · g∗)2]− 2g∗(20 · g∗)− 10 = 0
g∗ ≈ 4.76, G∗ ≈ 95
π∗ ≈ 4327
2.5.3 Social optimum
• The social optimal number of goats is
max
G
G · [10000−G2]− 10G
GSO ≈ 58, gSO = GSO/20 ≈ 3
πSO ≈ 19216
• In the unique Nash equilibrium, the green is over-grazed. Each farmer earns a signifi-
cantly lower profit than the social optimum.
• Why do farmers overgraze? When a farmer grazes an additional goat, he enjoys the
entire value of it, but only absorbs a share of the cost.
• Same stories today with environment, traffic congestion, shirk duties in a team project,
etc.
• Game theory: given the payoffs specified in this example and similar examples, the
only rational prediction is that people will overproduce/over-consume. This is just a
result of people being rational.
• How to fix it? Change people’s payoffs. E.g., penalty / reward (physical or moral)
3 Practice problems
1. Players 1 and 2 are fighting over how to split 100 dollars. Their friend passes by and
suggests the following rule:
Both players simultaneously name shares they would like to have,s1 and s2, which are
numbers between 0 and 100. If s1 + s2 ≤ 100, then the players receive the numbers
they named; if s1 + s2 > 100, then both players receive zero. (The friend gets the
left-over money.)
(a) Are the following strategy profiles Nash equilibrium?
(0, 100)
(0, 99)
(1, 100)
(50, 50)
(80, 20)
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(b) Find all pure-strategy Nash equilibria of this game.
2. In the following game, which strategy pairs survive the iterated elimination of strictly
dominated strategies? Which are the pure-strategy Nash equilibria?
Player 2
L M R
Player 1
U 2,1 3,1 3,5
M 6,4 3,3 3,1
D 4,3 6,0 4,1
3. Cournot competition with infinitely many homogeneous firms
There are n firms that produce homogenous goods. Each firm i chooses to produce at
quantity qi ≥ 0. The total cost to produce qi is Ci(qi) = 10qi for each firm i. When
the total quantity produced by all firms is Q = q1 + q2 + …+ qn, the market price for
the good is P (Q) = 100−Q.
(a) Find the Nash equilibrium of this game. Express optimal quantities as functions
of n.
(b) What is the market price in the Nash equilibrium? What is each firm’s profit?
How does the price and the profit change with n?
(c) Calculate the limit equilibrium price and profit as n converges to infinity.
4. Bertrand competition with heterogeneous firms
There are two firms that produce heterogeneous goods. Each firm i chooses to sell its
products at price i. The quantity that consumers demand from firm i is
qi(pi, pj) = 100− pi + pj
(Firm i’s good is a substitute for Firm j’s good, so an increase in pj makes more
consumers choose Firm i.)
There are no fixed costs of production and marginal costs are constant at 10. The
firms announce their prices simultaneously.
Find the Nash equilibrium in this game. Calculate the equilibrium profits for each
firm.
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