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Topic 5a

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Topic 5a

  1. 1. Managerial Economics: Topic 5 Sequential Games & Bargaining Sequential Games Commitments Commitments in Bargaining
  2. 2. Some History <ul><li>The tools of game theory are now common-place within economics. They were originally developed by John von Neumann and Oscar Morgenstern in their 1944 book, The Theory of Games and Economic Behavior . </li></ul><ul><li>Thomas Schelling in his 1956 book The Strategy of Conflict was the first to apply game theory to many contexts in social sciences. He won a Nobel Prize in 2005, along with Robert Aumann, another game theorist, for their work in addressing the question: why do some individuals/groups/countries succeed in cooperating, while others suffer from conflict? </li></ul><ul><li>The theory has developed to a high degree of mathematical sophistication. The importance of this development was signified by the award of the 1994 Nobel Prize to three game theorists: John Nash, John Harsanyi, and Reinhard Selten. </li></ul>
  3. 3. Topic 5a Introduction to Sequential Games Sequential Choices: Represented by games in extensive form (also called game trees)
  4. 4. Game Trees <ul><li>A game tree has choice nodes (squares) for each player, and chance nodes (circles) when Nature plays. </li></ul>Charlie Brown Accept Reject Lucy Pull ball away Let Charlie Brown kick
  5. 5. An Entry Game <ul><li>Vacuum cleaner market currently has one incumbent (Fastcleaners) </li></ul><ul><li>Potential entrant (Newcleaners). It is deciding whether to enter the market or not. </li></ul><ul><li>If enters, Fastcleaners has 2 choices: </li></ul><ul><ul><li>Accommodate: accept a lower market share </li></ul></ul><ul><ul><li>Price war </li></ul></ul>
  6. 6. Newcleaners Fastcleaners Enter Keep Out Accommodate Fight Price War $100,000 = TR - Entry Cost =$150,000 - $50,000 -$25,000 = $25,000 - $50,000 Newcleaner’s Payoffs $0 to Newcleaners
  7. 7. What should it do? <ul><li>Newcleaner needs to forecast Fastcleaner’s response </li></ul><ul><li>How does it do this? </li></ul><ul><ul><li>Put themselves in Fastcleaner’s shoes </li></ul></ul><ul><ul><li>Work out Fastcleaner’s payoffs </li></ul></ul>
  8. 8. Work Backwards Newcleaners Fastcleaners Enter Keep Out Accommodate Fight Price War N: $100,000 F: $150,000 N:-$25,000 F: $25,000 N: $0 F: $300,000
  9. 9. Work Backwards Newcleaners Fastcleaners Enter Keep Out Accommodate Fight Price War N: $100,000 F: $150,000 N:-$25,000 F: $25,000 N: $0 F: $300,000
  10. 10. Backwards Induction <ul><li>The predicted outcome of extensive form games can be found by solving the game by rollback: </li></ul><ul><ul><li>the decisionmaker at each node makes the decision that gives her the highest payoff </li></ul></ul><ul><ul><li>She doesn’t consider others’ payoffs, except to determine how others will act at their decision nodes. </li></ul></ul><ul><li>In the entry game, the solution is found by finding Fastcleaners’ best response to entry. If Newcleaners enters, Fastcleaners would rather accommodate entry than fight a price war. (A threat of entry is not credible.) So entry is worthwhile. </li></ul><ul><li>Sequential Games: In-class exercises </li></ul><ul><li>To notice for later: </li></ul><ul><li>Sequential games do not necessarily lead to a surplus-maximizing outcome! </li></ul>
  11. 11. <ul><li>Solving nodes with uncertainty is the same as before, except that there are now two payoffs to keep track of. </li></ul><ul><li>Should David Jones stock extra umbrellas this summer? </li></ul><ul><li>DJ </li></ul>Games with Nature as a player: Uncertainty Stocks extra umbrellas doesn’t NATURE Rains often 0.2 Drought 0.2 YOU YOU YOU buy don’t $10, $5 -$5, -$10 - $5, $0 -$5, - $1 $10, $2 $10, - $5 Rains less 0.6 $0, ?
  12. 12. Making Credible Threats: ex: controlling rotten kids <ul><li>Refuse to go </li></ul>Difficult Child Agree to go Punish Relent Parent (-1, -1) (2, 0) (1, 1)
  13. 13. <ul><li>Problem : The parent threatens the child with a punishment that is not credible , because it hurts both the parent and the child.  The child will refuse to go. </li></ul><ul><li> Unless the parent can commit to punish if the child refuses to go, (s)he cannot convince the child to go. </li></ul><ul><ul><li>Similarly, Lucy cannot commit not to pull away the ball  Charlie Brown should not kick </li></ul></ul><ul><li>What are some of the ways to commit in this case? </li></ul><ul><ul><li>reputation </li></ul></ul><ul><ul><li>delegation </li></ul></ul>Credibility
  14. 14. <ul><ul><li>What if the USA sets up an automatic nuclear response EARLY? </li></ul></ul>Example: Nuclear Deterrence USA Invade Europe Don’t invade Accommodate Fight Nuclear War (0 to USSR, 0 to USA) USSR (-100 to USSR, -100 to USA) (1 to USSR, -1 to USA)
  15. 15. <ul><li>Automatic response system is set up </li></ul><ul><li>“ Accommodate” branch is ruled out BEFORE </li></ul><ul><li>the game </li></ul>Nuclear Deterrence in Dr. Strangelove USA Invade Europe Don’t invade Accommodate Fight Nuclear War (0 to USSR, 0 to USA) USSR (-100 to USSR, -100 to USA) (1 to USSR, -1 to USA)
  16. 16. <ul><li>“ Accommodate” branch is ruled out BEFORE the game </li></ul><ul><li>USSR will not </li></ul><ul><li>invade </li></ul>Nuclear Deterrence in Dr. Strangelove USA Invade Europe Don’t invade Accommodate Fight Nuclear War (0 to USSR, 0 to USA) USSR (-100 to USSR, -100 to USA) (1 to USSR, -1 to USA)
  17. 17. Role of Commitments <ul><li>You commit to a future action that you would not normally want to take, in order to influence the choices of other players. (Preventing invasion,...) </li></ul><ul><li>(This is different from providing information to other players about what actions you’ll want to take: </li></ul><ul><ul><li>if you don’t study at all, and don’t come to class, you’ll fail = information </li></ul></ul><ul><ul><li>if you work hard, but still don’t understand at all, you’ll fail = commitment to a performance standard. </li></ul></ul>
  18. 18. Commitment Mechanisms (Dixit and Nalebuff, Thinking Strategically ) <ul><li>establishing a reputation </li></ul><ul><li>writing an enforceable contract </li></ul><ul><li>cutting off communication </li></ul><ul><li>burning bridges behind you </li></ul><ul><li>leaving the outcome out of your control </li></ul><ul><ul><ul><li>(3rd party arbitration, for example) </li></ul></ul></ul><ul><li>moving in small steps </li></ul><ul><li>developing credibility through teamwork </li></ul><ul><li>employing mandated negotiating agents </li></ul>
  19. 19. Application: Bargaining with a specific protocol <ul><li>Almost all bargaining is a series of offers and counter-offers </li></ul><ul><li>it’s a sequential game . </li></ul><ul><li>Example: The value of commitments </li></ul><ul><li>ArgoSoft is negotiating the acquisition of B-tech, another software company that is facing bankruptcy. </li></ul><ul><li>The negotiations drag on, and approach the deadline by which B-tech must enter bankruptcy proceeding. </li></ul><ul><li>The CEO of ArgoSoft sends a (signed!) acquisition agreement to B-tech’s lawyers, at the same time as he disappears for a vacation that will last until after the deadline, and is unreachable. </li></ul><ul><li>Suppose $20 million is ArgoSoft’s Willingness-to-Pay </li></ul><ul><li>(B-tech’s Willingness-to-Sell is probably $0) </li></ul>
  20. 20. The ultimatum game (one round) Argo B-tech If B-tech’s lawyers do not sign the agreement, each gets nothing. Offer $0.1 Offer $5 $10 Offer $15 Accept Reject A R R R A A A: $19.9 B: $0.1 A: $0 B: $0 $15, $5 $10, $10 $5, $15 $0, $0 $0, $0 $0, $0
  21. 21. Solve by rollback Argo B-tech If Argo offers $15m, B-tech’s best response is to accept. Offer $0.1 Offer $5 $10 Offer $15 Accept Reject A R R R A A A: $19.9 B: $0.1 $15, $5 $10, $10 $5, $15 $0, $0 $0, $0 $0, $0 A: $0 B: $0
  22. 22. Solve by rollback Argo B-tech So we can eliminate the ‘reject’ branch in that situation. Offer $0.1 Offer $5 $10 Offer $15 Accept Reject A R R R A A A: $19.9 B: $0.1 $15, $5 $10, $10 $5, $15 $0, $0 $0, $0 $0, $0 A: $0 B: $0
  23. 23. Solve by rollback Argo B-tech But the same holds for any other positive offer – so B-tech will always accept. Offer $0.1 Offer $5 $10 Offer $15 Accept Reject A R R R A A A: $19.9 B: $0.1 $15, $5 $10, $10 $5, $15 $0, $0 $0, $0 $0, $0 Remember – these are the complete payoffs and this is the complete game. A: $0 B: $0
  24. 24. Solve by roll back Argo So now the game is simple – the Argo CEO should offer B-tech the minimum amount and he will expect B-tech to accept. Offer $0.1 Offer $5 $10 Offer $15 Argo: $19.9 B-tech: $0.1 $5, $15 $10, $10 $15, $5
  25. 25. So:Argo gets almost all the surplus <ul><li>WHY? </li></ul><ul><ul><li>ArgoSoft gets to make a take-it-or-leave-it offer to B-tech, so it gets all the surplus . </li></ul></ul><ul><ul><li>Question : Why can’t Argo make a take-it-or-leave-it offer without cutting off communication? </li></ul></ul><ul><ul><li>So bargaining power depends on the bargaining process </li></ul></ul><ul><ul><li>If you can dictate/influence the bargaining process, it’s worth doing! </li></ul></ul><ul><ul><li>These tactics will damage long-term relationships  don’t try this with a regular business partner! </li></ul></ul>
  26. 26. Warning! Studies of our sense of fairness <ul><li>Fairness probably tends to make extreme results less likely : </li></ul><ul><li>The experiment is as follows: </li></ul><ul><ul><li>Subject 1 is given a sum of money and ask to divide it between herself and Subject 2. </li></ul></ul><ul><ul><li>Subject 1 makes a take-it-or-leave it offer of a “split” </li></ul></ul><ul><ul><li>Subject 2 has the option of accepting 1’s offer or rejecting it. </li></ul></ul><ul><ul><li>If accepted, they get 1’s proposed division; if rejected, they get nothing. </li></ul></ul><ul><ul><li>(E.g., Subject One proposes to divide $10 but keeping $7 and giving subject Two $3. If Two accepts, One gets $7 and Two gets $3, but if Two rejects they each get $0). </li></ul></ul><ul><li>Note that theory predicts that subject 1 should offer subject 2 only one cent, and subject 2 should accept. </li></ul>
  27. 27. Results: don’t push your luck! 100 11.8 % of 1’s demands > 90% 99+ 61.0 Avg % Demanded by 1 in Accepted Proposals 85.3 Avg % Demanded by 1 in Rejected Proposals 0 21.5 % rejected by 2 0 25.5 % of Proposed 50-50 splits 99+ 67.1 Avg % of Total Demanded by 1 Predicted Actual
  28. 28. 2 forces pushing results towards a fair split <ul><ul><li>Your ethics, sense of fairness </li></ul></ul><ul><ul><li>Risk that the other party get angry, will reject the offer even if it’s in their best interest to accept, for the satisfaction of lowering your profits. </li></ul></ul>
  29. 29. Another example of fixed protocol <ul><ul><li>Parent gives Ben and Jerry a tub of ice cream to share; they must agree on a split before they can have any ice cream. </li></ul></ul><ul><ul><li>Ben is oldest, so he gets to make the first offer; then Jerry, then Ben again,... </li></ul></ul><ul><ul><li>The ice cream is gradually melting while they’re arguing: every time an offer is rejected, a third of the ice cream melts before they can make another offer and reach agreement. </li></ul></ul><ul><li>In-Class Exercise: </li></ul><ul><ul><li>What will be the split, ignoring “fairness”? </li></ul></ul><ul><ul><li>How long does it take them to reach agreement? </li></ul></ul><ul><ul><li>(How would “fairness” affect the results?) </li></ul></ul>

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