Outline for Multi-Stage Games
Dr. Edward L. Millner
Fast Track MBA
Spring 2005
Learning Objectives:
Define a sequential game.
Identify Nash equilibria for sequential games.
Use backward induction to identify subgame perfect equilibria for sequential games.
Distinguish credible threats from those that are incredible.
Identify Nash equilibria for simple sequential entry games.
Identify actions that can shape simple sequential entry or product location games to a player’s advantage.
Integrate game theory with the strategic dilemmas.
Preparation
Read pp. 374-382 and 468-473 in Baye.
Review the notes that follow.
Jot down answers or thoughts for the questions in the notes. These answers will not be collected. However, spending a little time thinking about how you would approach a problem will increase your understanding and retention of the material. If you have no idea how to approach a problem, simply write “Lost” or something similar.
Agenda
Entry games
Backward induction
Credible and incredible threats
Subgame perfection
Shape the game
Product location games
Backward induction
Preemption
In a multi-stage game, one or more players move first and these moves are revealed to the other players before the other players choose their actions.
A strategy specifies the actions taken by a player for every possible contingency. Think of a strategy for a multi-stage game as a program telling a computer what to do at each decision node. By including a series of IF statements the choice at a node can depend upon the public choices made at all previous nodes. The IF statements must address every possible contingency! Otherwise, the strategy is incomplete.
Rule: Decide your response for every contingency before the game begins
Entry Game (p. 377)
Two players, A, the potential entrant and B, the incumbent
The potential entrant moves first and chooses between IN or OUT
OUT results in payoffs of (pA = 0, pB = 10)
If IN, then the incumbent chooses between HARD or SOFT
(IN, HARD) results in payoffs of (-1, 1)
(IN, SOFT) results in payoffs of (5, 5)
B, the incumbent, has two possible strategies from which to choose. What are they?
What is the normal form of the game?
How many Nash equilibrium exist for this game? What is it (are they)?
Rule: Use the extensive form of the game to determine the best strategy. That is, look ahead and reason back.
Using the extensive form (see p. 377), what payoff should A anticipate if it chooses IN? Given this anticipation, what strategy would you suggest for A? for B?
The “best” strategy identified by the extensive form and backward induction is sub-game perfect. Sub-game perfection means that playing a strategy is best for the player for every possible contingency that may arise. Since the payoff to B is higher with SOFT than with HARD if A chooses IN, (IN, HARD if IN) is not sub-game perfect.
Rule: Use sub-game perfection to distinguish between threats that are credible and those that are not credible. Carrying out a credible threat is in the threatener’s best interest. Carrying out an incredible threat is not in the threatener’s best interest.
Would your recommendation for A change if B took out a full page ad in the WSJ announcing plans to play HARD if A chooses IN?
Tipping Game: You are on a business trip in an American city that you will never visit again. You have just eaten in a nice restaurant and received excellent service from the staff. You do not know anyone at the restaurant and expect never to see anyone from there again. What is the sub-game perfect amount to tip?
Many players will not chose sub-game perfect strategies
Additional factors that seem to influence deviations from subgame perfection include:
Fairness
Reciprocity
Pareto dominance
Focal point
Communication
Standards and customs
These factors also help determine which Nash equilibrium tends to predominate when two or more exist
Miller-Busch game: Years ago Miller Brewing Co. announced their intention to become the largest brewer in the USA. Augie Busch responded, “Fine. Tell them to bring lots of money.”
What did Augie Busch mean?
Is his threat credible?
Repeated entry game: Suppose that you are the incumbent in 20 markets, each with a different potential entrant. You repeat the entry game in each market. That is, the potential entrant in the first market chooses IN or OUT, you announce your choice in that market, and payoffs are determined, announced, and distributed. The game is then repeated in the second market with the second potential entrant. This process is repeated for each market.
If the first potential entrant chooses IN, would you respond with HARD? Why or why not?
If the 20th potential entrant chooses IN, would you respond with HARD? Why or why not?
Entry game with pre-stage capacity choice
Two players, A, the potential entrant and B, the incumbent
The incumbent moves first and chooses between LARGE or SMALL
The potential entrant then and chooses between IN or OUT
(SMALL, OUT) results in payoffs of (pA = 0, pB = 10)
(LARGE, OUT) results in payoffs of (0, 7)
If the potential entrant chooses IN, the incumbent then chooses between HARD or SOFT.
(SMALL, IN, HARD) results in payoffs of (-1, 1)
(SMALL, IN, SOFT) results in payoffs of (5, 5)
(LARGE, IN, HARD) results in payoffs of (-5,-1)
(LARGE, IN, SOFT) results in payoffs of (3, -4)
If the incumbent chooses LARGE, is the threat to play HARD if IN credible?
Would you recommend that the incumbent play LARGE or SMALL?
What is the sub-game perfect equilibrium?
Rule: Look ahead and shape the game to your benefit.
Product location game (I)
The game has two players, A and B
A and B brew beer
Beer has two qualities: maltiness and hoppiness
Consumer tastes are uniformly distributed between the maltiness and hoppiness
A moves first and chooses where on the continuum between maltiness and hoppiness to locate its beer
B then chooses its product location
Each consumer purchases the beer closest to his or her taste (Ties are distributed evenly.)
Profit to a seller is a linear function of sales
What is your recommendation for A?
Product location game (II)
The game has three players, A, B, and C
A moves first, then B, then C
What is your recommendation for A?
Product location game (III)
The game has two players, A and B
A moves first and can produce an infinite variety of beers
B moves second and can also produce an infinite variety of beers
Each variety requires an “inch” on the maltiness-hoppiness continuum to break even
What is your recommendation for A?
Rule: Preempt competition when possible.
Time permitting, the following questions will be examined as in-class, group exercises.
SELLER and BUYER are engaged in a multi-stage game. BUYER decides whether to purchase (P) or lease (L) a crucial machine. Then SELLER decides whether to post a high price (H) or a low price (O). The game ends when BUYER decides whether to buy (B) or not (N). If SELLER purchases the machine, the payoff table is
Payoffs if BUYER purchases the machine |
|||
Player |
|
SELLER |
|
|
|
H |
O |
BUYER |
B |
-5, 35 |
5, 30 |
N |
-10, 0 |
-10, 0 |
If BUYER leases the machine and R is the rent paid to the SELLER, the payoff table is
Payoffs if BUYER leases the machine |
|||
Player |
|
SELLER |
|
|
|
H |
O |
BUYER |
B |
5 - R, 25 + R |
15 - R, 20 + R |
N |
0, 0 |
0, 0 |
If R = 3, what are the payoffs to SELLER and BUYER in the sub-game perfect Nash equilibrium? Does BUYER purchase or lease in this equilibrium? A brief explanation is optional.
If SELLER sets R at the same time as it chooses H or O and R is limited to integer values, what value of R would maximize its payoff in a sub-game perfect Nash equilibrium? A brief explanation is optional.
Suppose that by some quirk of nature, you are the only potential entrant into a market which is currently monopolized by one firm. However, you must enter by the end of the week or forever lose the opportunity. The incumbent firm’s annual cost function is given by CI = 10,000 + 10qI + 0.25*qI2, where qI is the incumbent’s annual output. You have a similar cost function for a homogeneous product. The inverse demand function is given by P = 500 - Q where Q denotes the total amount offered for sale in a year. The incumbent has learned of your interest and has increased production to 270 units and announced its intentions to maintain this level of output if you enter. If you do not enter by the end of the week, the firm will be guaranteed its monopoly position for an infinite number of years.
The following table shows the monopoly outcome, the Cournot outcome, and the best outcome for you if you enter and the incumbent maintains its current level of output.
|
QI |
QE |
P |
ΠI |
ΠE |
Monopoly |
196 |
0 |
304 |
38020 |
0 |
Cournot |
140 |
140 |
220 |
14500 |
14500 |
Best Response to 270 |
270 |
88 |
142 |
7415 |
-320 |
a. Would you enter the market? Explain fully, describing the price and sales volume that you anticipate in the market after your entry.
b. Would the incumbent support a regulation requiring it and all entrants (including you if you decide to enter) a licensing fee of 15,000?
Suppose that the Brazilian government awards two firms, Motorola and Sony, the exclusive rights to share the market for cellular service in Brazil. Motorola and Sony can both provide either analog or digital cellular phones. The payoff table below shows the annual profits each would earn, in millions, for the four possible combinations. The original contract called for both firms to enter the market simultaneously. However, the government is considering a proposal to delay one firm’s entry into the market for two years. The stated aim of the proposal is to prevent “destructive competition.”
|
|
Motorola |
|
|
|
Analog (MA) |
Digital (MD) |
Sony |
Analog (SA) |
10, 14 |
8, 9 |
Digital (SD) |
9.5, 11 |
12, 11.5 |
Four strategy combinations exist in a simultaneous move game: (SA, MA), (SA, MD), (SD, MA), and (SD, MD). Circle each combination that is a Nash equilibrium.
If the government delays Motorola’s entry and Sony moves first, is a threat by Motorola use Analog phones if Sony uses Digital phones a credible threat? Explain briefly.
If the government delays Motorola’s entry and Sony moves first, what profit outcomes would you predict for Motorola and Sony? Explain briefly?
If the government delays Sony’s entry and Motorola moves first, what profit outcomes would you predict for Motorola and Sony? Explain briefly?
If the discount rate is zero and the firm is risk neutral, how much would Motorola be willing to pay to be able to move first instead of second?
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