Nash Equilibrium:
Illustrations
Cournot’s Model of Oligopoly
• Single good produced by n firms
• Cost to firm i of producing qi units: Ci(qi), where
Ci is nonnegative and increasing
• If firms’ total output is Q then market price is
P(Q), where P is nonincreasing
• Profit of firm i, as a function of all the firms’
outputs:
 q , q
i
1
, ...,
2
q
n
 q
i
P
 n


 j 1
q

C
j

i
q 
i
Cournot’s Model of Oligopoly
Strategic game:
• players: firms
• each firm’s set of actions: set of all possible
outputs
• each firm’s preferences are represented by its
profit
Example: Duopoly
• two firms
• Inverse demand:
• constant unit cost: Ci(qi) = cqi, where c < a
Example: Duopoly
Example: Duopoly
Recall for a perfectly competitive firm, P=MC, so
a – Q = c, or Q = a – c.
Recall for a monopolist, MR=MC, so
a – 2Q = c, or Q = (a – c)/2.
Example: Duopoly
Example: Duopoly
• Best response function is:
Same for firm 2: b2(q) = b1(q) for all q.
Example: Duopoly
Example: Duopoly
Nash equilibrium:
Pair (q*1, q*2) of outputs such that each firm’s action
is a best response to the other firm’s action
or
q*1 = b1(q*2) and q*2 = b2(q*1)
Solution:
q1 = ( a - c - q2)/2 and q2 = (a - c - q1)/2
q*1 = q*2 = (a - c)/3
Example: Duopoly
Example: Duopoly
Conclusion:
• Game has unique Nash equilibrium:
(q*1, q*2) = (( a - c)/3, (a - c)/3)
• At equilibrium, each firm’s profit is
 = (a - c)2)/9
• Total output 2/3(a- c) lies between monopoly
output (a - c)/2 and competitive output a - c.
Cournot’s Model of Oligopoly
• Comparison of Nash equilibrium with collusive
outcomes. Can both firms do better than the
Cournot Nash? Yes! (if they can collude)
– Exercise 60.2 in Osborne
• Dependence of Nash equilibrium on number of
firms. What happens to the Nash quantity and
price as the number of firms grows?
– Exercise 61.1 in Osborne
Bertrand’s Model of Oligopoly
• Strategic variable price rather than output.
• Single good produced by n firms
• Cost to firm i of producing qi units: Ci(qi), where
Ci is nonnegative and increasing
• If price is p, demand is D(p)
• Consumers buy from firm with lowest price
• Firms produce what is demanded
Bertrand’s Model of Oligopoly
Strategic game:
• players: firms
• each firm’s set of actions: set of all possible prices
• each firm’s preferences are represented by its profit
Example: Duopoly
• 2 firms
• Ci(qi) = cqi for i = 1, 2
• D(p) = a - p
Example: Duopoly
Nash Equilibrium (p1, p2) = (c, c)
If each firm charges a price of c then the other firm
can do no better than charge a price of c also (if
it raises its price it sells no output, while if it
lowers its price it makes a loss), so (c, c) is a
Nash equilibrium.
Example: Duopoly
No other pair (p1, p2) is a Nash equilibrium since
• If pi < c then the firm whose price is lowest (or
either firm, if the prices are the same) can
increase its profit (to zero) by raising its price to c
• If pi = c and pj > c then firm i is better off
increasing its price slightly
• if pi ≥ pj > c then firm i can increase its profit by
lowering pi to some price between c and pj (e.g. to
slightly below pj if D(pj) > 0 or to pm if D(pj) = 0).
Example: Duopoly
p2
45
pm
P2 = BR2(p1)
P1 = BR1(p2)
c
c
pm
p1
Hotelling’s Model of Electoral
Competition
• Several candidates run for political office
• Each candidate chooses a policy position
• Each citizen, who has preferences over policy
positions, votes for one of the candidates
• Candidate who obtains the most votes wins.
Hotelling’s Model of Electoral
Competition
Strategic game:
• Players: candidates
• Set of actions of each candidate: set of possible
positions
• Each candidate gets the votes of all citizens who
prefer her position to the other candidates’
positions; each candidate prefers to win than to
tie than to lose.
Note: Citizens are not players in this game.
Example
• Two candidates
• Set of possible positions is a (one-dimensional)
interval.
• Each voter has a single favorite position, on
each side of which her distaste for other
positions increases equally.
• Unique median favorite position m among the
voters: the favorite positions of half of the voters
are at most m, and the favorite positions of the
other half of the voters are at least m.
Example
Direct argument for Nash equilibrium
(m, m) is an equilibrium: if either candidate chooses
a different position she loses.
No other pair of positions is a Nash equilibrium:
• If one candidate loses then she can do better by
moving to m (where she either wins or ties for first
place)
• If the candidates tie (because their positions are
either the same or symmetric about m), then
either candidate can do better by moving to m,
where she wins.
The War of Attrition
• Two parties involved in a costly dispute
• E.g. two animals fighting over prey
• Each animal chooses time at which it intends to
give up
• Once an animal has given up, the other obtains
all the prey
• If both animals give up at the same time then
they split the prey equally.
• Fighting is costly: each animal prefers as short a
fight as possible.
Also a model of bargaining between humans.
The War of Attrition
Strategic game
• players: the two parties
• each player’s set of actions is [0,∞) (set of possible
concession times)
• player i’s preferences are represented by payoff
function
The War of Attrition
• If t1 = t2 then either player can increase her
payoff by conceding slightly later (in which case
she obtains the object for sure, rather than
getting it with probability 1/2 ).
• If 0 < ti < tj then player i can increase her payoff
by conceding at 0.
• If 0 = ti < tj < vi then player i can increase her
payoff (from 0 to almost vi - tj > 0) by conceding
slightly after tj.
The War of Attrition
• Thus there is no Nash equilibrium in which
t1 = t2, 0 < ti < tj, or 0 = ti < tj < vi (for i = 1
and j = 2, or i = 2 and j = 1).
• The remaining possibility is that 0 = ti < tj
and tj ≥ vi for i = 1 and j = 2, or i = 2 and j =
1. In this case player i’s payoff is 0, while if
she concedes later her payoff is negative;
player j’s payoff is vj, her highest possible
payoff in the game.
The War of Attrition
• In no equilibrium is there any fight
• There is an equilibrium in which either
player concedes first, regardless of the
sizes of the valuations.
• Equilibria are asymmetric, even when v1 =
v2, in which case the game is symmetric.
Auctions
Second-Price Sealed-Bid Auction
Assume every bidder knows her own valuation and
every other bidder’s valuation for the good being
sold
Model
• each person decides, before auction begins,
maximum amount she is willing to bid
• person who bids most wins
• person who wins pays the second highest bid.
Second-Price Sealed-Bid Auction
Strategic game:
• players: bidders
• set of actions of each player: set of possible bids
(nonnegative numbers)
• preferences of player i: represented by a payoff
function that gives player i vi - p if she wins
(where vi is her valuation and p is the secondhighest bid) and 0 otherwise.
Second-Price Sealed-Bid Auction
Simple (but arbitrary) tie-breaking rule: number
players 1, . . . , n and make the winner the player
with the lowest number among those that submit
the highest bid.
Assume that v1 > v2 > · · · > vn.
Nash equilibria of second-price sealedbid auction
One Nash equilibrium
(b1, . . . , bn) = (v1, . . . , vn)
Outcome: player 1 obtains the object at price v2;
her payoff is v1 - v2 and every other player’s
payoff is zero.
Nash equilibria of second-price sealedbid auction
Reason:
• Player 1:
– If she changes her bid to some x ≥ b2 the
outcome does not change (remember she pays
the second highest bid)
– If she lowers her bid below b2 she loses and
gets a payoff of 0 (instead of v1 - b2 > 0).
Nash equilibria of second-price sealedbid auction
• Players 2, . . . , n:
– If she lowers her bid she still loses
– If she raises her bid to x ≤ b1 she still loses
– If she raises her bid above b1 she wins, but
gets a payoff vi - v1 < 0.
Nash equilibria of second-price sealedbid auction
Another Nash equilibrium
(v1, 0, . . . , 0) is also a Nash equilibrium
Outcome: player 1 obtains the object at price 0;
her payoff is v1 and every other player’s payoff is
zero.
Nash equilibria of second-price sealedbid auction
Reason:
• Player 1:
– any change in her bid has no effect on the
outcome
• Players 2, . . . , n:
– if she raises her bid to x  v1 she still loses
– if she raises her bid above v1 she wins, but
gets a negative payoff vi - v1.
Nash equilibria of second-price sealedbid auction
• For each player i the action vi weakly
dominates all her other actions
• That is: player i can do no better than bid
vi no matter what the other players bid.
Nash equilibria of second-price sealedbid auction
I
II
III
IV
Payoff if
b^-i  b’i
Payoff if
b’i < b^-i  vi
Payoff if
vi < b^-i  b’’i
Payoff if
b’’i < b^-i
b’i < vi
b’i = vi
b’’i > vi
vi - b^-i
vi - b^-i
vi - b^-i
+
0
+
vi - b^-i
+
vi - b^-i
+
0
0
+
vi - b^-i
-
0
0
0
First-Price Sealed-Bid Auction
Strategic game:
• players: bidders
• actions of each player: set of possible bids
(nonnegative numbers)
• preferences of player i: represented by a payoff
function that gives player i vi - p if she wins
(where vi is her valuation and p is her bid) and 0
otherwise.
Nash Equilibria of First-Price Sealed-Bid
Auction
(b1, . . . , bn) = (v2, v2, v3, . . . , vn) is a Nash
equilibrium
Reason:
• If player 1 raises her bid she still wins, but pays a
higher price and hence obtains a lower payoff. If
player 1 lowers her bid then she loses, and
obtains the payoff of 0.
• If any other player changes her bid to any price at
most equal to v2 the outcome does not change. If
she raises her bid above v2 she wins, but obtains
a negative payoff.
Nash Equilibria of First-Price SealedBid Auction
Property of all equilibria
• In all equilibria the object is obtained by the player
who values it most highly (player 1)
Argument:
• If player i ≠ 1 obtains the object then we must
have bi > b1.
• But there is no equilibrium in which bi > b1 :
• If bi > v2 then i’s payoff is negative, so she can do
better by reducing her bid to 0
• If bi ≤ v2 then player 1 can increase her payoff
from 0 to v1 - bi by bidding bi.
First-Price Sealed-Bid Auction
• As in a second-price auction, any player i’s
action of bidding bi > vi is weakly dominated by
the action of bidding vi :
– If the other players’ bids are such that player i
loses when she bids bi, then it makes no
difference to her whether she bids bi or vi
– If the other players’ bids are such that player i
wins when she bids bi, then she gets a
negative payoff bidding bi and a payoff of 0
when she bids vi
First-Price Sealed-Bid Auction
• Unlike a second-price auction, a bid bi < vi of
player i is NOT weakly dominated by any bid.
– It is not weakly dominated by a bid bi’ < bi
– Neither by a bid bi’ > bi
Revenue Equivalence
The price at which the object is sold, and
hence the auctioneer’s revenue, is the same
in the equilibrium (v1, v2, . . . , vn) of the
second-price auction as it is in the equilibrium
(v2, v2, v3, . . . , vn) of the first-price auction.