Game theory

Business Strategy in Oligopoly Markets

Introduction
In the majority of markets firms interact with few competitors
In determining strategy each firm has to consider rival’s reactions
strategic interaction in prices, outputs, advertising …
This kind of interaction is analyzed using game theory
assumes that “players” are rational
Distinguish cooperative and noncooperative games
focus on noncooperative games
Also consider timing
simultaneous versus sequential games

What is Game Theory ?
“ No man is an island”
Study of rational behavior in interactive or interdependent situations
Bad news:
Knowing game theory does not guarantee winning
Good news:
Framework for thinking about strategic interaction

Why Study Game Theory?
Because we can formulate effective strategy…
Because we can predict the outcome of strategic situations…
Because we can select or design the best game for us to be playing…

Definition of a Game
Must consider the strategic environment
Who are the PLAYERS? (Decision makers)
What STRATEGIES are available? (Feasible actions)
What are the PAYOFFS? (Objectives)
Rules of the game
What is the time-frame for decisions? Sequential or simultaneous?
What is the nature of the conflict?
What is the nature of interaction?
What information is available?

Oligopoly Theory
No single theory
employ game theoretic tools that are appropriate
outcome depends upon information available
Need a concept of equilibrium
players (firms?) choose strategies, one for each player
combination of strategies determines outcome
outcome determines pay-offs (profits?)
Equilibrium first formalized by Nash: No firm wants to change its current strategy given that no other firm changes its current strategy

Nash Equilibrium
Equilibrium need not be “nice”
firms might do better by coordinating but such coordination may not be possible (or legal)
Some strategies can be eliminated on occasions
they are never good strategies no matter what the rivals do
These are dominated strategies
they are never employed and so can be eliminated
elimination of a dominated strategy may result in another being dominated: it also can be eliminated
One strategy might always be chosen no matter what the rivals do: dominant strategy

An Example
Two airlines
Prices set: compete in departure times
70% of consumers prefer evening departure, 30% prefer morning departure
If the airlines choose the same departure times they share the market equally
Pay-offs to the airlines are determined by market shares
Represent the pay-offs in a pay-off matrix

The example (cont.) The Pay-Off Matrix American Delta Morning Morning Evening Evening (15, 15) The left-hand number is the pay-off to Delta The right-hand number is the pay-off to American (30, 70) (70, 30) (35, 35) What is the equilibrium for this game?

The example (cont.) The Pay-Off Matrix American Delta Morning Morning Evening Evening (15, 15) If American chooses a morning departure, Delta will choose evening (30, 70) (70, 30) (35, 35) If American chooses an evening departure, Delta will still choose evening The morning departure is a dominated strategy for Delta and so can be eliminated. The Nash Equilibrium must therefore be one in which both airlines choose an evening departure (35, 35) The morning departure is also a dominated strategy for American and again can be eliminated

Dominant Strategy
A strategy that outperforms all other choices
no matter what opposing players do
COMMANDMENT:
If you have a dominant strategy, use it.
Expect your opponent to use her dominant strategy if she has one.

The example (cont.)
Now suppose that Delta has a frequent flier program
When both airline choose the same departure times Delta gets 60% of the travelers
This changes the pay-off matrix

The example (cont.) The Pay-Off Matrix American Delta Morning Morning Evening Evening (18, 12) (30, 70) (70, 30) (42, 28) However, a morning departure is still a dominated strategy for Delta (Evening is still a dominant strategy. If Delta chooses a morning departure, American will choose evening But if Delta chooses an evening departure, American will choose morning American has no dominated strategy American knows this and so chooses a morning departure (70, 30)

Successive Deletion of Dominated Strategies
Rational players …
Should play dominant strategies
Should not play dominated strategies
Should not expect others to play dominated strategies
 Thus, dominated strategies may be eliminated from consideration
This may be done iteratively

Example: Tourists & Natives
Two bars (bar 1, bar 2) compete
Can charge price of $2, $4, or $5
6000 tourists pick a bar randomly
4000 natives select the lowest price bar
Example 1: Both charge $2
each gets 5,000 customers
Example 2: Bar 1 charges $4,
Bar 2 charges $5
Bar 1 gets 3000+4000=7,000 customers
Bar 2 gets 3000 customers

Tourists & Natives Bar 2 25 , 25 28 , 15 14 , 15 $5 $4 15 , 28 15 , 14 $5 20 , 20 12 , 14 $4 Bar 1 14 , 12 10 , 10 $2 $2

Successive Elimination of Dominated Strategies
Does any player have a dominant strategy?
Does any player have a dominated strategy?
Eliminate the dominated strategies
Reduce the normal-form game
Iterate the above procedure
What is the equilibrium?

Successive Elimination of Dominated Strategies 25 , 25 28 , 15 14 , 15 $5 $4 15 , 28 15 , 14 $5 20 , 20 12 , 14 $4 Bar 1 14 , 12 10 , 10 $2 $2 , , , , , , , Bar 1 , , Bar 2 Bar 2 25 , 25 28 , 15 $5 $4 15 , 28 $5 20 , 20 $4 Bar 1

Nash Equilibrium
What if there are no dominated or dominant strategies?
The Nash equilibrium concept can still help us in eliminating at least some outcomes
Nash Equilibrium :
A set of strategies, one for each player, such that each player’s strategy is best for her given that all other players are playing their equilibrium strategies
Best Response :
The best strategy I can play given the strategy choices of all other players
Nash equilibrium: Everybody is playing a best response
No incentive to unilaterally change my strategy

Example:
Change the airline game to a pricing game:
60 potential passengers with a reservation price of $500
120 additional passengers with a reservation price of $220
price discrimination is not possible (perhaps for regulatory reasons or because the airlines don’t know the passenger types)
costs are $200 per passenger no matter when the plane leaves
the airlines must choose between a price of $500 and a price of $220
if equal prices are charged the passengers are evenly shared
Otherwise the low-price airline gets all the passengers
The pay-off matrix is now:

The example (cont.) The Pay-Off Matrix American Delta P H = $500 ($9000,$9000) ($0, $3600) ($3600, $0) ($1800, $1800) P H = $500 P L = $220 P L = $220 If both price high then both get 30 passengers. Profit per passenger is $300 If Delta prices high and American low then American gets all 180 passengers. Profit per passenger is $20 If Delta prices low and American high then Delta gets all 180 passengers. Profit per passenger is $20 If both price low they each get 90 passengers. Profit per passenger is $20

Nash Equilibrium (cont.) The Pay-Off Matrix American Delta P H = $500 ($9000,$9000) ($0, $3600) ($3600, $0) ($1800, $1800) P H = $500 P L = $220 P L = $220 (P H , P L ) cannot be a Nash equilibrium. If American prices low then Delta should also price low ($0, $3600) (P L , P H ) cannot be a Nash equilibrium. If American prices high then Delta should also price high ($3600, $0) (P H , P H ) is a Nash equilibrium. If both are pricing high then neither wants to change ($9000, $9000) (P L , P L ) is a Nash equilibrium. If both are pricing low then neither wants to change ($1800, $1800) There are two Nash equilibria to this version of the game There is no simple way to choose between these equilibria. But even so, the Nash concept has eliminated half of the outcomes as equilibria Custom and familiarity might lead both to price high “ Regret” might cause both to price low

Nash Equilibrium (cont.) The Pay-Off Matrix American Delta P H = $500 ($9000,$9000) ($0, $3600) ($3600, $0) ($1800, $1800) P H = $500 P L = $220 P L = $220 (P H , P L ) cannot be a Nash equilibrium. If American prices low then Delta would want to price low, too. ($0, $3600) (P L , P H ) cannot be a Nash equilibrium. If American prices high then Delta should also price high ($3600, $0) (P H , P H ) is a Nash equilibrium. If both are pricing high then neither wants to change ($9000, $9000) (P L , P L ) is a Nash equilibrium. If both are pricing low then neither wants to change ($1800, $1800) There are two Nash equilibria to this version of the game There is no simple way to choose between these equilibria, but at least we have eliminated half of the outcomes as possible equilibria

Nash Equilibrium (cont.) The Pay-Off Matrix American Delta P H = $500 ($9000,$9000) ($0, $3600) ($3600, $0) ($1800, $1800) P H = $500 P L = $220 P L = $220 ($0, $3600) ($3600, $0) ($3,000, $3,000) ($1800, $1800) Delta can see that if it sets a high price, then American will do best by also pricing high. Delta earns $9000 Suppose that Delta can set its price first Delta can also see that if it sets a low price, American will do best by pricing low. Delta will then earn $1800 The only sensible choice for Delta is P H knowing that American will follow with P H and each will earn $9000. So, the Nash equilibria now is (P H , P H ) ($1800, $1800) Sometimes, consideration of the timing of moves can help us find the equilibrium This means that P H , P L cannot be an equilibrium outcome This means that P L ,P H cannot be an equilibrium

Game theory

by Alistercrowe

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