Game theory

727 views
692 views

Published on

0 Comments
1 Like
Statistics
Notes
  • Be the first to comment

No Downloads
Views
Total views
727
On SlideShare
0
From Embeds
0
Number of Embeds
0
Actions
Shares
0
Downloads
21
Comments
0
Likes
1
Embeds 0
No embeds

No notes for slide

Game theory

  1. 1. Business Strategy in Oligopoly Markets
  2. 3. Introduction <ul><li>In the majority of markets firms interact with few competitors </li></ul><ul><li>In determining strategy each firm has to consider rival’s reactions </li></ul><ul><ul><li>strategic interaction in prices, outputs, advertising … </li></ul></ul><ul><li>This kind of interaction is analyzed using game theory </li></ul><ul><ul><li>assumes that “players” are rational </li></ul></ul><ul><li>Distinguish cooperative and noncooperative games </li></ul><ul><ul><li>focus on noncooperative games </li></ul></ul><ul><li>Also consider timing </li></ul><ul><ul><li>simultaneous versus sequential games </li></ul></ul>
  3. 4. What is Game Theory ? <ul><li>“ No man is an island” </li></ul><ul><li>Study of rational behavior in interactive or interdependent situations </li></ul><ul><li>Bad news: </li></ul><ul><li>Knowing game theory does not guarantee winning </li></ul><ul><li>Good news: </li></ul><ul><li>Framework for thinking about strategic interaction </li></ul>
  4. 5. Why Study Game Theory? <ul><ul><li>Because we can formulate effective strategy… </li></ul></ul><ul><ul><li>Because we can predict the outcome of strategic situations… </li></ul></ul><ul><ul><li>Because we can select or design the best game for us to be playing… </li></ul></ul>
  5. 6. Definition of a Game <ul><li>Must consider the strategic environment </li></ul><ul><ul><ul><li>Who are the PLAYERS? (Decision makers) </li></ul></ul></ul><ul><ul><ul><li>What STRATEGIES are available? (Feasible actions) </li></ul></ul></ul><ul><ul><ul><li>What are the PAYOFFS? (Objectives) </li></ul></ul></ul><ul><li>Rules of the game </li></ul><ul><ul><ul><li>What is the time-frame for decisions? Sequential or simultaneous? </li></ul></ul></ul><ul><ul><ul><li>What is the nature of the conflict? </li></ul></ul></ul><ul><ul><ul><li>What is the nature of interaction? </li></ul></ul></ul><ul><ul><ul><li>What information is available? </li></ul></ul></ul>
  6. 7. Oligopoly Theory <ul><li>No single theory </li></ul><ul><ul><li>employ game theoretic tools that are appropriate </li></ul></ul><ul><ul><li>outcome depends upon information available </li></ul></ul><ul><li>Need a concept of equilibrium </li></ul><ul><ul><li>players (firms?) choose strategies, one for each player </li></ul></ul><ul><ul><li>combination of strategies determines outcome </li></ul></ul><ul><ul><li>outcome determines pay-offs (profits?) </li></ul></ul><ul><li>Equilibrium first formalized by Nash: No firm wants to change its current strategy given that no other firm changes its current strategy </li></ul>
  7. 8. Nash Equilibrium <ul><li>Equilibrium need not be “nice” </li></ul><ul><ul><li>firms might do better by coordinating but such coordination may not be possible (or legal) </li></ul></ul><ul><li>Some strategies can be eliminated on occasions </li></ul><ul><ul><li>they are never good strategies no matter what the rivals do </li></ul></ul><ul><li>These are dominated strategies </li></ul><ul><ul><li>they are never employed and so can be eliminated </li></ul></ul><ul><ul><li>elimination of a dominated strategy may result in another being dominated: it also can be eliminated </li></ul></ul><ul><li>One strategy might always be chosen no matter what the rivals do: dominant strategy </li></ul>
  8. 9. An Example <ul><li>Two airlines </li></ul><ul><li>Prices set: compete in departure times </li></ul><ul><li>70% of consumers prefer evening departure, 30% prefer morning departure </li></ul><ul><li>If the airlines choose the same departure times they share the market equally </li></ul><ul><li>Pay-offs to the airlines are determined by market shares </li></ul><ul><li>Represent the pay-offs in a pay-off matrix </li></ul>
  9. 10. 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?
  10. 11. 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
  11. 12. Dominant Strategy <ul><li>A strategy that outperforms all other choices </li></ul><ul><li>no matter what opposing players do </li></ul><ul><li>COMMANDMENT: </li></ul><ul><li>If you have a dominant strategy, use it. </li></ul><ul><li>Expect your opponent to use her dominant strategy if she has one. </li></ul>
  12. 13. The example (cont.) <ul><li>Now suppose that Delta has a frequent flier program </li></ul><ul><li>When both airline choose the same departure times Delta gets 60% of the travelers </li></ul><ul><li>This changes the pay-off matrix </li></ul>
  13. 14. 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)
  14. 15. Successive Deletion of Dominated Strategies <ul><li>Rational players … </li></ul><ul><ul><li>Should play dominant strategies </li></ul></ul><ul><ul><li>Should not play dominated strategies </li></ul></ul><ul><ul><li>Should not expect others to play dominated strategies </li></ul></ul><ul><ul><li> Thus, dominated strategies may be eliminated from consideration </li></ul></ul><ul><ul><li>This may be done iteratively </li></ul></ul>
  15. 16. Example: Tourists & Natives <ul><li>Two bars (bar 1, bar 2) compete </li></ul><ul><li>Can charge price of $2, $4, or $5 </li></ul><ul><li>6000 tourists pick a bar randomly </li></ul><ul><li>4000 natives select the lowest price bar </li></ul><ul><li>Example 1: Both charge $2 </li></ul><ul><ul><li>each gets 5,000 customers </li></ul></ul><ul><li>Example 2: Bar 1 charges $4, </li></ul><ul><li>Bar 2 charges $5 </li></ul><ul><ul><li>Bar 1 gets 3000+4000=7,000 customers </li></ul></ul><ul><ul><li>Bar 2 gets 3000 customers </li></ul></ul>
  16. 17. 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
  17. 18. Successive Elimination of Dominated Strategies <ul><li>Does any player have a dominant strategy? </li></ul><ul><li>Does any player have a dominated strategy? </li></ul><ul><ul><ul><li>Eliminate the dominated strategies </li></ul></ul></ul><ul><ul><ul><li>Reduce the normal-form game </li></ul></ul></ul><ul><ul><ul><li>Iterate the above procedure </li></ul></ul></ul><ul><li>What is the equilibrium? </li></ul>
  18. 19. 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
  19. 20. Nash Equilibrium <ul><li>What if there are no dominated or dominant strategies? </li></ul><ul><li>The Nash equilibrium concept can still help us in eliminating at least some outcomes </li></ul><ul><li>Nash Equilibrium : </li></ul><ul><li>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 </li></ul><ul><li>Best Response : </li></ul><ul><ul><ul><li>The best strategy I can play given the strategy choices of all other players </li></ul></ul></ul><ul><li>Nash equilibrium: Everybody is playing a best response </li></ul><ul><ul><ul><li>No incentive to unilaterally change my strategy </li></ul></ul></ul>
  20. 21. Example: <ul><li>Change the airline game to a pricing game: </li></ul><ul><ul><li>60 potential passengers with a reservation price of $500 </li></ul></ul><ul><ul><li>120 additional passengers with a reservation price of $220 </li></ul></ul><ul><ul><li>price discrimination is not possible (perhaps for regulatory reasons or because the airlines don’t know the passenger types) </li></ul></ul><ul><ul><li>costs are $200 per passenger no matter when the plane leaves </li></ul></ul><ul><ul><li>the airlines must choose between a price of $500 and a price of $220 </li></ul></ul><ul><ul><li>if equal prices are charged the passengers are evenly shared </li></ul></ul><ul><ul><li>Otherwise the low-price airline gets all the passengers </li></ul></ul><ul><li>The pay-off matrix is now: </li></ul>
  21. 22. 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
  22. 23. 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
  23. 24. 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
  24. 25. 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

×