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Game Theory
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  1. Game Theory Topic 3 Sequential Games “It is true that life must be understood backward, but … it must be lived forward.” - Søren Kierkegaard

  2. Review • Understanding the outcomes of games • Sometimes easy • Dominant strategies • Sometimes more challenging • “I know that you know …” • What if a game is sequential? • Market entry

  3. Very Large Airplanes:Airbus vs. Boeing • Airbus • “ The problem is the monopoly of the 747 … They have a product. We have none. ” - Airbus Executive • Initiated plans to build a super-jumbo jet • Industry feasibility studies: • Room for at most one megaseater • Boeing • “Boeing, the world’s top aircraft maker, announced it was building a plane with 600 to 800 seats, the biggest and most expensive airliner ever.” - BusinessWeek

  4. out out in in Sequential Games The Game $0, $0 out – $1 billion, – $1billion Airbus $0.3 billion, – $3 billion in Boeing – $4 billion, – $4billion

  5. out in Looking Forward … • Airbus makes the first move: • Must consider how Boeing will respond • If stay out: • Boeing stays out $0billion Boeing – $1 billion

  6. out in Looking Forward … • Airbus makes the first move: • Must consider how Boeing will respond • If enter: • Boeing accommodates, stays out – $3billion Boeing – $4 billion

  7. out in … And Reasoning Back • Now consider the first move: • Only ( In, Out ) is sequentially rational • In is not credible (for Boeing) $0, $0 out Airbus $0.3 billion, – $3 billion out Boeing

  8. out out in in What if Boeing Can Profit? The Game $0, $0 out – $1 billion, + $1billion Airbus $0.3 billion, – $3 billion in Boeing – $4 billion, – $4billion

  9. Nash Equilibria Are Deceiving • Two equilibria (game of chicken) • But, still only one is sequentially rational

  10. Airbus vs. Boeing • October 2007 • A380 enters commercial service • Singapore to Sydney • List price: $350 million • September 2011 • Four year anniversary: 12,000,000 seats sold

  11. Solving Sequential Games • Intuitive Approach: • Start at the end and trim the tree to the present • Eliminates non-credible future action

  12. Solving Sequential Games • Steps: • What action will each last player to act take? • Look only at that player’s payoffs • Select the highest • Place an arrow on the selected branch • Delete all other branches • Now, treat the next-to-last player to act as last • Continue in this manner until you reach the root • Equilibrium: the “name” of each arrow

  13. Subgame Perfect Equilibrium • Subgame: • A decision node and all nodes that follow it • Subgame Perfect Equilibrium: (a.k.a. Rollback, Backwards Induction) • The equilibrium specifies an action at every decision node in the game • The equilibrium is also an equilibrium in every subgame

  14. Nash Equilibria Are Deceiving • Does either player have a dominant strategy? • What is the equilibrium? • What if Player 1 goes first? • What if Player 2 goes first?

  15. Solving Sequential Games • Thinking backwards is easy in game trees • Start at the end and trim the tree to the present • Thinking backwards is challenging in practice • Outline: • Strategic moves in early rounds • The rule of three (again) • Seeing the end of the game

  16. Graduation SpeakerRevisited Graduation speaker Ron Paul, Sarah Palin, or Al Gore? • Four committee members prefer: Ron to Sarah to Al ( R > S > A ) • Three committee members prefer: Sarah to Al to Ron ( S > A > R ) • Two committee members prefer: Al to Ron to Sarah ( A > R > S ) • Voting by Majority Rule

  17. Graduation SpeakerRevisited Graduation speaker Ron Paul, Sarah Palin, or Al Gore? • Member preferences: 4 (R>S>A) 3 (S>A>R) 2 (A>R>S) • Majority rule results: • R beats S ; S beats A ; A beats R • Voting results (example): • R beats S then winner versus AA

  18. Voting as a Sequential Game R R vs. A A R R vs. S S S A S vs. A

  19. Looking Forward … B R R vs. A A majority prefers A to R A R S S A majority prefers S to A A S vs. A A

  20. … And Reasoning Back Three committee members prefer R to S to A. How should they vote in the first round? R vs. A A R A R vs. S S S S S vs. A

  21. Sequential Rationality Look forward and reason back. Anticipate what your rivals will do tomorrow in response to your actions today

  22. Importance of Rules • Outcome is still predetermined: • R beats S then winner versus A • R beats A then winner versus S • S beats A then winner versus R

  23. Accommodating a Potential Entrant • Do you enter? • Do you accommodate entry?

  24. Survivor Immunity Challenge • There are 21 flags • Players alternate removing 1, 2, or 3 flags • The player to take the last flag wins

  25. Unraveling give give give give 6, 6 take take take take 2, 1 2, 4 6, 3 3, 9

  26. Unraveling • Equilibrium: take , take ; take , take • Remember: • An equilibrium specifies an action at every decision node • Even those that will not be reached in equilibrium

  27. Sequential Games • You have a monopoly market in every state • There is one potential entrant in each state • They make their entry decisions sequentially • Florida may enter today • New York may enter tomorrow • etc. • Each time, you can accommodate or fight • What do you do the first year?

  28. E3 E2 out out E1 fight fight M in in M acc acc The Game

  29. out $0, $100 + previous E in 50, 50 + previous acc M fight –50, –50 + previous Looking Forward … • In the last period: • No reason to fight final entrant, thus ( In, Accommodate )

  30. … And Reasoning Back • The Incumbent will not fight the last entrant • But then, no reason to fight the previous entrant • … • But then, no reason to fight the first entrant • Only one sequential equilibrium • All entrants play In • Incumbent plays Accommodate • But for long games, this is mostly theoretical • People “see” the end two to three periods out!

  31. Breakfast Cereals A small sampling of the Kellogg’s portfolio

  32. Breakfast Cereals product development costs: $1.2M per product 600 500 400 sales (in thousands) 300 200 100 000 1 2 3 4 5 6 7 8 9 10 11 less sweet more sweet

  33. Breakfast Cereals

  34. Breakfast Cereals 600 500 400 sales (in thousands) 300 200 100 000 1 2 3 4 5 6 7 8 9 10 11 less sweet more sweet

  35. First Product Entry SCENARIO 1 Profit = ½ 5(600) – 1200 = 300 600 500 400 sales (in thousands) 300 200 100 000 1 2 3 4 5 6 7 8 9 10 11 less sweet more sweet

  36. Second Product Entry SCENARIO 2 Profit = 2 x 300 = 600 600 500 400 sales (in thousands) 300 200 100 000 1 2 3 4 5 6 7 8 9 10 11 less sweet more sweet

  37. Third Product Entry SCENARIO 3 Profit = 300 x 3 – 240 x 2 = 420 600 500 400 sales (in thousands) 300 200 100 000 1 2 3 4 5 6 7 8 9 10 11 less sweet more sweet

  38. Competitor Enters SCENARIO 4 Profit = 300 x 2 - 240 = 360 600 500 400 sales (in thousands) 300 200 100 000 1 2 3 4 5 6 7 8 9 10 11 less sweet more sweet

  39. Strategic Voting • We saw that voting strategically rather than honestly can change outcomes • Other examples? • Amendments to make bad bills worse • Crossing over in open primaries • “Centrist” voting in primaries

  40. Strategic Voting • Maybe majority rule causes this. • Can we eliminate “strategic voting” with other rules? • Ranking of all candidates • Proportional representation • Run offs • Etc.

  41. Arrow’s Impossibility Theorem • Consider a voting rule that satisfies: • If everyone prefers A to B, B can’t win • If A beats B and C, then A beats B • The only political procedure that always guarantees the above is a dictator • No voting system avoids strategic voting

  42. Summary • Thinking forward misses opportunities • Make sure to see the game through to the logical end • Don’t expect others to see the end until it is close • The rule of three steps