1. Teoria dei giochi e Oligopolio
2014/15

2. Game theory Game theory Oligopolistic firm use a
a set of tools used to analyze the decision making by firms
Oligopolistic firm use a
strategy: battle plan of actions (such as setting a price or quantity) to compete with other firms
Oligopolies engage in a
game: competition between players (firms) in which strategic behavior plays a major role

3. Nash equilibrium (NE)
A set of strategies is a “Nash equilibrium” (NE) if,
holding strategies of all other players (firms) constant,
no player (firm) can obtain a higher payoff (profit) by choosing a different strategy
In a Nash equilibrium, no firm wants to change its strategy because each firm is using its “best response” (strategy that maximizes its profit given its beliefs about its rivals' strategies)

4. Example (NE)
Consider a duopoly, single-period and quantity-setting game
American Airlines (A) and
United Airlines (U)
Q = total number of passengers flown by both firms; sum of:
qA = passengers on American Airlines
qU = passengers on United Airlines

5. Firms act SIMULTANEOUSLY
Each firm selects a strategy that
maximizes its profit
given what it believes other firm will do
Firms are playing
a “non-cooperative” game under IMPERFECT INFORMATION: each firm must choose an action before observing rivals’ simultaneous actions

7. Dominant strategy (DS)
DS: a strategy that strictly dominates all other strategies, regardless the action taken by the rival
in this game, each firm has a dominant strategy
each firm chooses its dominant strategy
where a firm has a dominant strategy, its belief about its rival's behavior is irrelevant

8. Non-cooperative game firms do not cooperate in a single-period game
In Nash equilibrium (qA = qU = 64), each firm earns $4.1 million (< $4.6 million they would obtain restricting their outputs to qA = qU = 48)
sum of firms' profits is NOT maximized in this simultaneous choice, one-period game

9. Why don't firms cooperate?
each firm has a substantial PROFIT INCENTIVE TO CHEAT on a collusive agreement
don't cooperate due to a LACK OF TRUST:
each firm can profitably use low-output strategy only if it trusts other firm

10. Prisoners' dilemma game
All players have dominant strategies that lead to a profit (or other payoff) that is inferior to what they could achieve if they cooperated and played alternative strategies B
“If you both confess, you will serve 2 years in prison; if only one confesses, he will go free, while the other will serve 5 years in prison; if you both do not confess, you serve 1 year in prison”  What is the final outcome? Both the robbers confess!
Confess
Not confess
-2;-2
0;-5
-5;0
-1;-1 A

11. American Airlines’ Profit-Maximizing Output
(a) Monopoly
(b) Duopoly p
, $ per p
, $ per
pass.er
pass.er
339
339
275
243
211 MC
147 MC
147 q
= 64 U D MR MR r D r D 96
169.5
339 64
128
137.5
275
339 q
, Thousand American Airlines q
, Thousand American Airlines A A
passengers per quarter
passengers per quarter

12. The best-response curve
The best-response curve shows the output each firm picks to maximize its profit, given its belief about its rival’s output
qU=0  qA=96
qU=64  qA=64
and so on

13. American and United’s Best-Response Curves
The CE is a set of quantities sold by firms such that, holding quantities of all other firms constant, no firm can obtain a higher profit by choosing a different quantity q
, Thousand United U
passengers per quarter
192
American ’
s best-response curve 96
Cournot equilibrium (CE) 64 48
United ’
s best-response curve 64 96
192 q
, Thousand American A
passengers per quarter

14. Duopoly Equilibria If the game is repeated over many period?
The possible cartel equilibria
lie on the contract curve
(a) Equilibrium Quantities q
, Thousand United U
passengers per quarter
192
American ’
s best-response curve
Contract
curve
Price-taking equilibrium 96
Cournot equilibrium 64
Stackelberg equilibrium 48
Cartel
equilibrium
United ’
s best-response curve 48 64 96
192 q
, Thousand American passengers per quarter A

15. Duopoly Equilibria
The highest possible Π for the two firms is given by the profit possibility frontier p
, $ million profit U
of United Airlines
9.2
Profit possibility frontier
Cartel profits
4.6
4.1
Cournot profits
2.3
Stackelberg
profits
American monopoly
Price-taking profits
profit
4.1
4.6
9.2 p
, $ million profit of American Airlines A

16. Collusion in repeated games
why, then, are cartels frequently observed?
collusion is more likely in a multiperiod game: single-period game played repeatedly
PUNISHMENT: not possible in a single-period game but possible in a multiperiod game

17. Supergame a single-period game played repeatedly
supergame: players’ strategies in each period may depend on rivals' actions in previous periods
in a repeated game, firm can influence its rival's behavior by
SIGNALING (using a low-quantity strategy for a couple of periods to signal to the other it is cooperative)
THREATENING to punish (for example, announcing the trigger strategy …)

18. Threat (the “trigger” strategy)
Suppose American announces to United that it will use the following two-part strategy:
American produces smaller quantity each period as long as United does the same
if United produces larger quantity in period (t), then American will produce larger quantity in period (t + 1) and (t + 2)
thus, if firms play same game indefinitely, they should find it more profitable to collude (unless the FIRM DOES NOT CARE ABOUT FUTURE PROFITS…)
Trigger strategy, an equilibrium of supergame

19. Number of periods (T)
suppose firms know that they are going to play game for T periods
period T is like a single-period game, and all firms cheat
by same reasoning, they cheat in T-1 period, (given no retaliation is possible in period T)
 maintaining a cartel is difficult if the game has a KNOWN STOPPING POINT
cheating is less likely to occur if end period is unknown or there is no end

20. Non-cooperative oligopoly
many models of non-cooperative oligopoly behavior
firms choose quantities
Cournot model (simultaneously)
Stackelberg model (two period game)

21. Stackelberg model
Cournot model: both firms make their output decisions simultaneously
Stackelberg's model: firms act sequentially
leader firm sets its output first
then its rival (follower) sets its output
The leader realizes that once it sets output, the follower will use its Cournot best-response curve to pick a “best-response” output

22. Stackelberg Equilibrium If AA sets q=192,
UA will set qUA=0, so D and DrAA are identical at q=192
If AA sets q=0, UA will set
q=96, so the DrAA curve is 96 less than demand. The DrAA hits the vertical axis at p=243
When AA sets qAA=96, UA sets qUA=48, …
(a) Residual Demand American Faces p
, $ per
passenger
339
243 D r
195 MR r
147 MC q
= 48 U D q
= 96 Q
= 144
192
339 A q
, Thousand American passengers per quarter
(b) United ’
s Best-Response Curve A q
, Thousand United U
passengers per quarter 96 q
= 48 U
United ’
s best-response curve q
= 96
192 A q
, Thousand American passengers per quarter A

23. Stackelberg Game Tree Leader ’ s decision Follower ’ s decision
Profits ( p , p ) A 48 U
(4.6, 4.6) 48 64
United
(3.8, 5.1) 96
(2.3, 4.6) 48
(5.1, 3.8) 64 64
American
United
(4.1, 4.1) 96
(2.0, 3.1) 48
(4.6, 2.3) 96 64
United
(3.1, 2.0) 96
(0, 0)

24. Stackelberg’s conclusion
If one firm (the Stackelberg leader) chooses the output before its rivals (the followers), the former (the leader) produces more and earns a higher profit than each (identical-cost) follower firms
Stackelberg equilibrium:
Q= 144 ( )
Cournot equilibrium:
Q= 128 ( )
Collusion equilibrium:
Q= 96 ( )

25. Question when firms move simultaneously, there is no leader
“But why doesn't AA ANNOUNCE it will produce Stackelberg-leader output, so as to induce UA to produce the Stackelberg follower's output level?”

26. No credible threat
when firms move SIMULTANEOUSLY, UA doesn't view AA's warning (that it will produce a large quantity) as a credible threat:
not in AA’s best interest to produce large quantity (i.e. qAA=96) because AA cannot be sure that UA believes threat and reduce its output, AA produces Cournot level (i.e. qAA=64)
when one firm moves FIRST, its threat to produce large quantity is credible because it has already COMMITTED to producing large quantity

27. Tabella di sintesi QA
A annuncia ? U
crede ? QU 96 c 48 nc 64

28. CASE: SIMULTANEOUS decisions
2 firms are considering opening a gas station at a highway rest stop
physical space for:
Scenario A: at most 2 gas stations (enough demand for 2 firms)
Scenario B: only 1 gas station (demand only for 1 firm)

29. firms have pure (dominant) strategies: both enter
unique, pure strategy equilibrium
neither firm has a dominant strategy
Sometimes, games do not have NE in pure strategies; sometimes there are two (or more) NE in mixed strategies

30. Problem with pure strategies
game has two Nash equilibria in pure strategies:
Firm 1 enters and Firm 2 does not
Firm 2 enters and Firm 1 does not
these pure Nash equilibria are “unappealing” because identical firms use different strategies
players don’t know which Nash equilibrium will result

31. Mixed strategies
mixed strategies: firm chooses between its possible actions with given probabilities
in our game, each firm enters with 50% probability
if strategies are mixed, firms may use same strategies
 result: Nash equilibrium in mixed strategies

32. Mixed strategy equilibria
if both firms use this mixed strategy, each of four outcomes in payoff matrix equally likely
Firm 1 has
¼ chance of earning $1 (upper-right cell)
¼ chance of losing $1 (lower-right cell)
½ chance of earning $0 (left cells)
thus, Firm 1's expected profit is
($1  ¼) + (-1  ¼) + (0  ½) = $0

33. Firm 2’s response
if Firm 1 uses this mixed strategy, Firm 2 cannot do better using a pure strategy
if Firm 2 enters with certainty, it earns
$1 half of the time
loses $1 the other half
so its expected profit is $0
if Firm 2 stays out with certainty, it earns $0 with certainty

34. Nash equilibria
One firm plays pure strategy of entering and other firm plays pure strategy of not entering or
Both firms play mixed strategies

35. Preventing entry: Sequential decisions
incumbent (monopoly) firm knows potential entrant is considering entering
stage 1: incumbent decides whether to take an action to prevent entry
stage 2: potential entrant decides whether to enter, and firms choose output levels
no entry: incumbent earns monopoly profit
entry: each firm earns duopoly profit
assume potential entrant enters only if it can make a positive profit

36. Incumbent: To act or not to act?
incumbent can act strategically to prevent other firm from entering
But, it has to answer the following question:
DOES IT PAY TO TAKE ACTION?
Three possibilities

37. Three possibilities blockaded entry: deterred entry:
market conditions make “profitable entry” impossible, so no action is necessary
deterred entry:
incumbent acts to prevent an additional firm from entering because it pays to be the only one
accommodated entry:
doesn't pay for incumbent to prevent entry
incumbent does nothing to prevent entry
reduces its output (or price) from monopoly to duopoly level

38. Two-stage game Example: Highways rest stop gas station
an INCUMBENT and a potential entrant
stage 1: incumbent decides whether to pay landlord (of the rest stop) b for exclusive right to be the only gas station
stage 2: if incumbent doesn't take this strategic action, potential entrant decides whether or not to enter

39. Whether an Incumbent Pays to Prevent Entry
First stage
Second stage ( π , π ) i e
Do not enter ( π
, $0) m
Do not pay
Entrant
Enter ( π , π = R – F ) d d
Incumbent
Pay for exclusive rights (entry is impossible) ( π – b
, $0) m

40. Incumbent’s decision
blockaded entry: duopoly profit is negative, πd < 0 (entry doesn't pay)
deterred or accommodated entry: πd > 0 (it means that entry occurs unless incumbent acts)
Does it deter or accommodate entry?
incumbent can prevent entry by paying b, but it may not pay for incumbent to do so

41. πm – b > πd Does incumbent pay? πd > πm - b
incumbent pays b if monopoly profit minus payment is greater than duopoly profit
πm – b > πd
incumbent accommodates entry if
πd > πm - b

42. Potential entrant: Fixed costs and demand
entry is profitable only if πd > 0
duopoly profit πd = R – F
assume firms have no variable costs
fixed cost of entering, F
firm’s revenue is R (depends on demand)

43. Blockaded entry
if R < F, πd < 0, and the second firm does not enter
not enough demand given fixed cost
entry is blockaded only if a firm must incur a fixed cost to enter
F = R > F πd > 0
F > 0
and demand so low there's room for only one firm in market ( a natural monopoly )

44. Incumbent’s advantage
Because incumbent is already in market,
its fixed entry cost is sunk
so it ignores its sunk cost in deciding whether to operate
potential entrant
views fixed cost of entry as avoidable cost
incurs cost only if entry takes place

45. The “credible threat”: Rationality and commitment
In the first place, rivals must believe that firm's threatened strategy is “rational”
rational: it is in firm's best interest to use it to prevent entry
In the second place, the “commitment” plays a crucial role:
the general “burning bridges” behind the army (it can only advance and not retreat)

46. Cournot vs. Stackelberg
Stackelberg models illustrate the role of “commitment”
Stackelberg model
leader chooses its output level before follower, so it has the first-mover advantage
moving first allows leader to COMMIT to producing a relatively large quantity, qs
Cournot model
firms choose output levels simultaneously
so no firm has an advantage over its rival
no firm can commit credibly to produce large quantity

47. Output commitment
incumbent can commit to “large quantity of output” before potential entrant decides whether to enter (sequential game)
3 possibilities:
NO commitment: entry occurs, COURNOT equilibrium
commit to STACKELBERG-LEADER quantity: entry occurs, Stackelberg equilibrium
commit to LARGER quantity: deters entry, monopoly equilibrium

48. Commitment and fixed cost
Does incumbent commit to producing the Stackelberg leader q or the larger entry-deterring q?
To answer, backward induction
What affects the incumbent decision? The potential entrant's fixed cost of entry (F)
If the F is below the level where entry is blockaded, there are three possibilities :
if it can’t commit  Cournot eq.
if it can commit:
F relatively high  Deterred-entry eq.
F relatively low  Stackelberg eq.

49. Game Trees for the Deterred Entry and Stackelberg Equilibria
(a) Entrant ’
s Fixed Cost Is $100. ( p , p ) i e
Do not enter
($900, $0)
Accommodate ( q
= 30) i
Entrant
Enter
($450, $125)
Incumbent
Do not enter
($800, $0)
Deter ( q
= 40) i
Entrant
Enter
($400, $0)
(b) Entrant ’
s Fixed Cost Is $16.
Do not enter
($900, $0)
Accommodate ( q
= 30) i
Entrant
Enter
($450, $209)
Incumbent
Do not enter
($416, $0)
Deter ( q
= 52) i
Entrant
Enter
($208, $0)

51. Stackelberg Equilibria
Cournot and
Stackelberg Equilibria
(a) Best-Response Curves q e
, Units
per period 60
Incumbent ’
s best-response curve 30 e c 20 e
F = $0; Stackelberg equilibrium
The incumbent profit is maximized at q=30, the Stackelberg equilibrium s 15
Entrant ’ s
best-response curve 20 30 60 q
, Units per period
(b) Incumbent ’
s Profit i p
, $ per period i
450
400 p i 20 30 60 q
, Units per period i

52. (a) Entrant ’
s Best-Response Curve 30
Entrant ’
s best-response curve
qe, Units per period e 15 s e 10 d 30 40 60 q
, Units per period
(b) Incumbent ’
s Profit
900
800
Incumbent Commits to a Large Quantity to Deter Entry F = 100$ The incumbent profit is maximized at q=40, where it deters entry and is a monopoly π m π
πi, Incumbent’s Profit per period, $ i
450 π s 30 40 60 q
, Units per period

53. Incumbent Loss If It Deters Entry F = $16 The incumbent profit
(a) Entrant ’
s Best-Response Curve
Incumbent
Loss If It Deters Entry q
, Units per period e 30
Entrant ’
s best-response curve e 15 s 30 52 60
F = $16
The incumbent profit
is maximized producing the
Stackelberg output (q=30),
instead of deterring entry by
producing q=52 q
, Units per period i
(b) Incumbent ’
s Profit p
, $ per period i
900 p m
450
416 p i p s 30 52 60 q
, Units per period i