1. normal form games with complete information
part 2

2. iterated elimination of dominated strategies dominance solvability
roadmap
iterated elimination of dominated strategies
dominance solvability
strong and weak versions
examples
discussion
rationality
order of elimination
non-existence

3. dominated strategies to sum up
si’ is strictly dominated by si for player i
if for all s-iє S-i, ui(si,s-i) > u i(si’,s-i)
si’ is (weakly) dominated by si for player i
if for all s-iє S-i, ui(si,s-i) ≥ u i(si’,s-i)
and for some s-iє S-i, ui(si,s-i) > u i(si’,s-i)
(and si is an undominated strategy for player i if it is not dominated by any other strategy)

4. example: prisoners’ dilemma 2 1 -1,-1 -6,0 0,-6 -3,-3 Not to confess

5. example: weakly dominated strategies2 1 L R U
1,1
0,1 M
0,2
1,0 D
0,-1
0,0

6. best-response and question
si is a best response to s-i
if for all s’iє Si, ui(si,s-i) ≥ u i(si’,s-i)
Can you relate the best-response concept with the concept of dominant strategy?
R: si is a dominant strategy iff it is a best-response to every s-i

7. iterated elimination of dominated strategies (IEDS)
If no rational player will play a dominated strategy, a rational player would not expect her opponents to play a dominated strategy
Elimination of dominated strategies can therefore lead to a chain reaction that successively narrows down how a group of rational players will act. If there is a unique prediction that arises from IEDS, that is the IEDS solution and the game is said to be dominance solvable
IEDS can be simultaneous or sequential

8. example of IEDS: weak 2 1 L M R U
1,1
0,1
1,2 D
-1,-1
-1,0
0,0

9. example of IEDS: strong2 1 L M R U
1,1
0,1
1,2 D
-1,-1
-1,0
0,0

10. iterated elimination of dominated strategies (IEDS)
A rational player never chooses a strictly dominated strategy
But a rational player may choose a dominated strategy!
Rationality is not enough to rule out weakly dominated stratgeies!
IE(W)DS requires rationality + knowing that others are rational + … (rationality has to be common knowledge) + knowing others’ payoffs.
Elimination of a strictly dominated strategy requires only rationality of the player.

11. discussion: order of elimination in IE(W)DS2 1 L R U
0,0
0,1 D
1,0
Order of elimintion matters bc leads to different outcomes!
Order of elimination matters in IEWDS when we remove dominated strategies in turn (ie, one player at a time). But not if we remove all players’ dominated strategies at the same time

12. discussion of IEDS: layers of rationality2 1 L C R U
4,5
1,6
5,6 M
3,5
2,5
5,4 D
2,0
7,0
R dominada por C
D dominada por M e U
L dominada por C
U dominada por M
Note-se que 2 pode garantir um payoff 5 jogando L; só joga C pq tem a certeza q 1 é racional…
1 nãousa D pq 2 é racional e não usa R
2 não usa L pq sabe que 1 é racional e sabe que 1 sabe que ele prórpio é racional: 2 não usa R, logo 1 não usa D, log 2 não usa L
1 não usa U pq sabe que 2 é racional e sabe que 2 sabe que ele próprio é racional e sabe que 2 sabe que ele sabe que 2 é racional….
Uff…
IE(W)DS requires rationality + knowing that others are rational + … (rationality has to be common knowledge) + knowing others’ payoffs.
Elimination of a strictly dominated strategy requires only rationality of the player.

13. discussion of IEDS: non-existence2 1
Heads
Tails
-1,+1
+1,-1

14. example 1 (homework)
Each player i of n players selects a number xi between 0 and 100 simultaneously.
Player i’s payoff is ui = xi-3y/5, where y denotes the average of all numbers chosen

15. example 1 (homework) Normal form representation
Is there any dominated strategy?
What numbers should be selected?
All strategies are strictly dominated by 100 (the utility function is strictly increasing in xi)
Everyone will select 100!

16. example 2 (homework)
Two players (1 and 2) are bidding at a painting’s auction.
Their valuations (represented by vi) (willingness to pay) are common knowledge and are such that v1 > v2.
The auction format is as follows. Each player i (simultaneously) submits a non-negative bid xi in a closed envelope. The auctioneer opens the envelopes. The highest bidder wins the auction and pays the other bidder’s bid. In case of a tie, bidder 1 wins the auction and pays the bid.

17. example 2 (homework) Normal form representation
Is there any dominated strategy?
What bids should be selected?

18. more homework:
exercises 1 and 2

19. normal form games with complete information
part 3

20. roadmap Nash equilibrium definition interpretations examples
relation between Nash equilibrium and IEDS
references
Sec 1.1.C and 1.2.A, B of Gibbons
Sec of Osborne
Ch 5 of Dutta

21. definition of Nash equilibrium
(recall) strategy si* is a best response to s-i*є S-i
if ui(si*,s-i*) ≥ ui(si,s-i*) for all si
A vector of strategies s*=(s1*,s2*,…,sN*) is
a Nash equilibrium
if ui(si*,s-i*) ≥ u i(si,s-i*) for all si and for all i

22. definition of Nash equilibrium
or si* solves
Max ui(s1*,..., si-1*, si, s-i+1*,..., sn*)
subject to si є Si
therefore, a Nash equilibrium is a profile of strategies, such that each player’s strategy is the best for her, given that the other players are playing equilibrium strategies

23. definition of Nash equilibrium
So,
Each player is playing a best response against a conjecture
Conjectures are correct

24. interpretations of Nash equilibrium
play prescription
preplay communication
rational introspection
focal point
trial and error
Play prescr: if the vector is proposed, then no-one wants to deviate
Preplay comm: players could meet before the game and coordinate on a NE
RI: each player asks himselfwhat the outcome will be; only NE appear to make sense
Focal: NE is a focal point (Schelling) bc it stands out
TE: NE is reached by trial and error… (!)

25. prisoners’ dilemma 2 1
Not to confess
Confess
-1,-1
-6,0
0,-6
-3,-3

26. coordination game problem: uniqueness2 1
Book launch
Movie
2,1
0,0
1,2

27. matching pennies problem: existence2 1
Heads
Tails
-1,+1
+1,-1

28. IE(W)DS and Nash equilibria2 1 L M R U
1,1
0,1
1,2 D
-1,-1
-1,0
0,0

29. IE(S)DS and Nash equilibria2 1 L M R U
1,1
0,1
1,2 D
-1,-1
-1,0
0,0

30. discussion: order of elimination, IE(W)DS and Nash equilibria2 1 L R U
0,0
0,1 D
1,0
When more than a NE exists, some may be eliminated by IEWDS

31. to sum up
Any IE(W)DS solution (i.e., when the game is dominance solvable) has to be a Nash equilibrium; there may be Nash equilibria that are not IE(W)DS solutions
Any IE(S)DS solution is a Nash equilibrium (but the strategies that survive IESDS don’t have to be a Nash equilibrium); all Nash equilibria survive IE(S)DS
IEWDS solvable implies NE; the reverse does not hold bc (1) IEWDS eliminates some NE when more than one NE exists and (2) the game may not be dominance solvable (eg, battle of sexes)

32. to sum up
A dominant strategy equilibrium must be a Nash equilibrium; the reverse implication does not hold

33. normal form games with complete information
part 4

34. roadmap solving a maximization problem Cournot model
Bertrand models (with differentiated and homogeneous products)
contributing to a public good
references
Sec 1.2.B of Gibbons and Sec 3.1 and 3.2 of Osborne
Ch 6 of Dutta

35. Cournot model of duopoly
a product is produced by only two firms: firm 1 and firm 2. The quantities are denoted by q1 and q2, respectively. Each firm chooses the quantity without knowing the other firm has chosen
the market price is P(Q)=a-Q, where a is a constant number and Q=q1+q2
the cost to firm i of producing quantity qi is
Ci (qi)=cqi

36. Cournot model of duopoly
how to find a Nash equilibrium:
find a quantity pair (q1*, q2*) such that q1* solves:
Max u1(q1, q2*)=q1[a-(q1+q2*)-c]
subject to 0 ≤ q1 ≤ +∞
and q2* solves
Max u2(q1*, q2)=q2[a-(q1*+q2)-c]
subject to 0 ≤ q2 ≤ +∞

37. concave function a function f(x) is concave if, for any x and y,
f(tx + (1- t)y) ≥ t f(x) + (1- t) f(y), for all 0 ≤ t ≤1

38. convex function a function f(x) is convex if, for any x and y,
f(tx + (1- t)y) ≤ t f(x) + (1- t) f(y), for all 0 ≤ t ≤1

39. concavity and convexity
a (differentiable) function f(x) is concave iff
f’’(x) ≤ 0
a (differentiable) function f(x) is convex iff
f’’(x) ≥ 0
A function f(x) is concave iff -f (x) is convex

40. concavity and convexity
example 1: f(x) = -3x2 + 6x - 4, f’(x) = -6x + 6, f’’(x) = -6 < 0. Hence, f(x) is concave
example 2: f(x) = ex, f’(x)= f’’(x) = ex > 0. Hence, f (x) is convex
example 3: f(x,y) = 2xy + 4y - x2 + 3y2
For any fixed y, f (x, y) is concave in x because
f’x(x,y) = 2y - 2x and f’’(x,y) = -2 < 0
For any fixed x, f (x, y) is convex in y because f’y(x,y) = 2x y and f’’y(x,y) = 6 > 0

41. maximum and minimum
a maximum of f(x) is a point x’ such that f(x’) ≥ f(x) for any x
a minimum of f(x) is a point x* such that f(x*) ≤ f(x) for any x

42. maximum and minimum
if a point x is a maximum or a minimum, then it satisfies the first order condition (FOC) f’(x) = 0
if f(x) is concave and x’ satisfies the FOC, then x’ is a maximum
if f(x) is convex and x* satisfies the FOC, then x* is a minimum

43. maximum and minimum
a maximum of f(x) in the domain [x1, x2] is a point x’ in [x1, x2] such that f(x’) ≥ f(x) for all x € [x1, x2]
a minimum of f(x) in the domain [x1, x2] is a point x* in [x1, x2] such that f(x*) ≤ f(x) for all x € [x1, x2]

44. finding a maximum of a concave function in [x1, x2]
find a maximum x’ of f(x) without constraints
if x’ is in [x1, x2] , then x’ is also a maximum for the constrained problem
otherwise,
if f(x1) > f( x2) or x’ < x1, then x1 is a maximum
if f(x1) < f( x2) or x’ > x2, then x2 is a maximum
if f(x1) = f( x2), then any point in [x1, x2] is a maximum

45. finding a maximum of a concave function in [x1, x2]
example 4:
Max f(x) = -3x2 + 6x – 4
subject to -2 ≤ x ≤ 2
solution: x = 1
example 5:
subject to x ≥ 2

46. finding a maximum of a concave function in [x1, x2]

47. Cournot model of duopoly
solve
Max u1(q1, q2*)=q1[a-(q1+q2*)-c]
subject to 0 ≤ q1 ≤ +∞
FOC: a -2q1 – q2* - c = 0
q1 = (a - q2* - c)/2
Max u2(q1*, q2)=q2[a-(q1*+q2)-c]
subject to 0 ≤ q2 ≤ +∞
FOC: a -2q2 – q1* - c = 0
q2 = (a – q1* - c)/2

48. Cournot model of duopoly
(q1*, q2*) is a Nash equilibrium iff
q1* = (a - q2* - c)/2
q2* = (a – q1* - c)/2
solving the two equations
q1* = q2*= (a - c)/2

49. Cournot model of duopoly
best response functions
R1(q2) = (a – q2 - c)/2 if q2 < a-c
0 otherwise
R2(q1) = (a – q1 - c)/2 if q1 < a-c

50. Cournot model of oligopoly
solve
Max ui(q1*,…, qi,…, qn*)=qi[a-(q1*+…+ qi+…+ qn*)-c]
subject to 0 ≤ qi ≤ +∞ …
FOC: a -2qi -(q1*+…+ qi-1+qi+1+…+ qn*)- c = 0
qi = (1)
Since all firms are symmetric, in NE we will have qi*= qj* for all i, j. Substituting in (1), we obtain qi*= (a-c)/(n+1) for all i. Therefore, NE is (q1*,…, qn*)=((a-c)/(n+1),…, (a-c)/(n+1))

51. application: the problem of the commons
n farmers in a village; each summer, all the farmers graze their goats on the village green
gi: number of goats owned by farmer I
c: cost of buying and caring for each goat
v: value of a goat is v(G) per goat, where
G = g1 + g gn and v(G): v’(G) < 0 and v’’(G) < 0
there is a maximum number of goats that can be grazed on the green: v(G)>0 if G < Gmax, and v(G)=0 if G ≥ Gmax
each spring, all the farmers simultaneously choose how many goats to own

52. application: the problem of the commons normal-form representation
set of players {farmer 1,…, farmer n}
sets of strategies Si = [0,Gmax), for i = 1, 2,…, n
payoff functions ui (g1,...,gn ) = gi v(g1+…+ gn ) – c gi for i =1, 2,…, n

53. application: the problem of the commons computing the Nash equilibrium
Max ui(g1*,…, gi-1*, gi, gi+1*,…, gn*)=
gi v(g1*+…+ gi-1*+ gi + gi+1*+…+ gn*) - cgi
subject to 0 ≤ gi ≤ Gmax
for i = 1, 2,…, n
FOC:
v(g1+ g2*+…+ gn*) + g1v’(g1+ g2 *+…+ gn*) – c = 0
v(g1*+ g2+…+ gn*) + g2v’(g1*+ g2 +…+ gn*) – c = 0 …
v(g1*+ g2*+…+ gn) + gnv’(g1*+ g2 *+…+ gn) – c = 0

54. application: the problem of the commons computing the Nash equilibrium
Summing over all farmers’ FOC’s and dividing by n yields
v(G*) + (1/n)G*v’(G*) – c = 0
where G*= g1*+ g2*+…+ gn*

55. application: the problem of the commons the social planner’s problem
Max Gv(G) – Gc
subject to 0 ≤ G ≤ Gmax
FOC
v(G) + Gv’(G) – c = 0
so, G**: v(G**) + G**v’(G**) – c = 0
G* vs G**?

56. to solve:
all exercises except for ex. 3 and 7