Buyer-Seller Game
BUYER-SELLER GAME
A buyer and a seller simultaneously submit a price, which can be any non-negative number.
Trade takes place when the price pb chosen by the buyer is at least high as the price ps chosen by the seller, i.e., pb ps
Saltuk Ozerturk (SMU)
Normal Form, Strict Dominance and Nash Equilibrium
Buyer-Seller Game
BUYER-SELLER GAME (continued)
If trade occurs, payo↵ of the buyer is his value v = 10 minus the price pb he pays and payo↵ of seller is the price pb she receives minus her cost c = 5.
If trade does not occur, both players receive zero payo↵.
Saltuk Ozerturk (SMU)
Normal Form, Strict Dominance and Nash Equilibrium
Buyer-Seller Game
Payo↵ to Buyer and Seller
ub(pb,ps)=(10 pb ifpb ps 0 otherwise.
us(pb,ps)=(pb 5 ifpb ps 0 otherwise.
Saltuk Ozerturk (SMU)
Normal Form, Strict Dominance and Nash Equilibrium
Buyer-Seller Game
Problem
Is the strategy profile pb = 7 and ps = 6 a NE. Explain formally. Solution
No it is not a NE, since
ub(7,6) = 3 < ub(6,6) = 4
Hencepb =7isnotaBRtops =6.
Saltuk Ozerturk (SMU)
Normal Form, Strict Dominance and Nash Equilibrium
Buyer-Seller Game (continued)
Problem
Is the strategy profile pb = 6 and ps = 6 a NE. Explain formally. Solution
Yes it is a NE since
Hence pb = 6 is a BR to ps = 6. Furthermore,
Henceps =6isaBRtopb =6.
ub(6,6) = 4
ub (6 + ", 6) = 4 "
ub (6 ", 6) = 0
us(6,6) = 1
us (6, 6 + ") = 0
us (6, 6 ") = 1
Saltuk Ozerturk (SMU)
Normal Form, Strict Dominance and Nash Equilibrium
Buyer-Seller Game (continued)
Problem
Are there any NE where pb = pb = x where 5 x 10? Solution
ub (x , x ) = 10 x 0 Hence pb = x is a BR to ps = x. Furthermore,
Henceps =x isaBRtopb =x.Therefore,anypb =pb =x where 5 x 10 is a NE
ub (x + ", x ) = 10 x "
ub (x ", x ) = 0
us (x , x ) = x 5 0
us (x , x + ") = 0
us (x , x ") = x 5
Saltuk Ozerturk (SMU)
Normal Form, Strict Dominance and Nash Equilibrium
Gunslingers
Consider the following duel. Each of two gunslingers independently chooses when to draw their weapon and fire.
The one who fires first wins the duel, which has a value 50, while the other loses, which has a cost 10.
Saltuk Ozerturk (SMU)
Normal Form, Strict Dominance and Nash Equilibrium
Gunslingers continued
However, if the one who draws first does so before 10 seconds has passed he su↵ers a reputation cost of 40.
If they fire at the same time, they win with equal probabilities.
Assume that the players can choose any nonnegative real number as their strategy.
Saltuk Ozerturk (SMU)
Normal Form, Strict Dominance and Nash Equilibrium
Gunslingers
Denote P1’s strategy as t1 and denote P2’s strategy as t2.
8> 50 >< 20
u1(t1, t2) = > 10 >: 20 10
if10t1
if 10 t2 < t1 if10t1 =t2
ift2 <10andt2
8><> 50 u2(t1,t2)= 10
10 Saltuk Ozerturk (SMU)
> 20 >: 20
Normal Form, Strict Dominance and Nash Equilibrium
Gunslingers
Problem
Is t1 = 30 and t2 = 30 a Nash Equilibrium? Explain formally. Solution
Note that
Hence this is not a NE.
u1(30, 30) = 20 < u1(29, 30) = 50
Saltuk Ozerturk (SMU)
Normal Form, Strict Dominance and Nash Equilibrium
Gunslingers
Problem
Are there any NE where t1 = t2 < 10? Explain formally. Solution
Consider any profile t1 = t2 = a < 10. We have u1(a, a) = 20 < u1(a ", a) = 10
Hence there is no such NE.
Saltuk Ozerturk (SMU)
Normal Form, Strict Dominance and Nash Equilibrium
Gunslingers
Problem
Are there any NE where 10 < t1 < t2? Explain formally. Solution
Consideranyprofile10