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

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 2 Mike Shor

Very Large Airplanes: Airbus vs. Boeing Industry background  “The problem is the monopoly of the 747 … They have a product. We have none. ” - Airbus Executive Industry feasibility studies:  Room for at most one megaseater Airbus  Initiated plans to build a super-jumbo jet 3 Mike Shor

Very Large Airplanes: Airbus vs. Boeing Boeing reaction “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 Mike Shor

Sequential Games Mike Shor 5 The Game Airbus Boeing – $4 billion, – $4 billion $0.3 billion, – $3 billion out in out in – $1 billion, – $1 billion $0, $0 out in

Looking Forward … Airbus makes the first move:  Must consider how Boeing will respond If stay out: Boeing stays out 6 Boeing $0 billion – $1 billion out in Mike Shor

Looking Forward … Airbus makes the first move:  Must consider how Boeing will respond If enter: Boeing accommodates, stays out 7 Boeing – $3 billion – $4 billion out in Mike Shor

… And Reasoning Back Now consider the first move: Only ( In, Out ) is sequentially rational  In is not credible (for Boeing) 8 Airbus Boeing $0, $0 out in out $0.3 billion, – $3 billion out Mike Shor

What if Boeing Can Profit? Mike Shor 9 The Game Airbus Boeing – $4 billion, – $4 billion $0.3 billion, – $3 billion out in out in – $1 billion, + $1 billion $0, $0 out in  ?

Nash Equilibria Are Deceiving Mike Shor 10 Boeing OutIn Airbus Out 0, 0 -1, 1 In 0.3, -3 -4, -4 Two equilibria (game of chicken) But, still only one is sequentially rational

Airbus vs. Boeing Mike Shor 11  October 2007  A380 enters commercial service  Singapore to Sydney  List price: $350 million  September 2011  Four year anniversary: 12,000,000 seats sold

Solving Sequential Games Intuitive Approach:  Start at the end and trim the tree to the present  Eliminates non-credible future actions Mike Shor 12

Solving Sequential Games Steps: 1. Pick a last move 2. What player is making the decision? 3. What decisions are available to that player? 4. What are that player’s payoffs from each decision? 5. Select the highest 6. Place an arrow on the selected branch 7. 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 Mike Shor 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 Mike Shor 14

Nash Equilibria Are Deceiving Mike Shor 15 Player 2 XY Player 1 Less 10, 0 30, 30 More 20, 20 40, 10 Does either player have a dominant strategy? What is the equilibrium? What if Player 1 goes first? What if Player 2 goes first?

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 Mike Shor 16

Graduation Speaker Revisited Graduation speaker Rand Paul, Jeb Bush, or Hilary Clinton? Four committee members prefer: Randto Jebto Hilary( R > J > H ) Three committee members prefer: Jebto Hilaryto Rand( J > H > R ) Two committee members prefer: Hilaryto Randto Jeb( H > R > J ) V oting by Majority Rule Mike Shor 17

Graduation Speaker Revisited Graduation speaker Rand Paul, Jeb Bush, or Hilary Clinton? Member preferences: 4 (R>J>H) 3 (J>H>R) 2 (H>R>J) Majority rule results:  R beats J ; J beats H ; H beats R Voting results (example):  R beats J then winner versus H  H Mike Shor 18

Voting as a Sequential Game Mike Shor 19 R vs. J R vs. H J vs. H R J R J H H R H J H

Looking Forward … Mike Shor 20 A majority prefers H to R A majority prefers J to H R vs. H J vs. H R J H H R H H J

… And Reasoning Back Four committee members prefer R to J to H. How should they vote in the first round? Mike Shor 21 R vs. J R vs. H J vs. H R J J H H J

Sequential Rationality Mike Shor 22 Look forward and reason back. Anticipate what your rivals will do tomorrow in response to your actions today

Importance of Rules Outcome is still predetermined:  R vs. J then winner versus H   J vs. H then winner versus R   H vs. R then winner versus J  Mike Shor 23

Accommodating a Potential Entrant Do you enter? Do you accommodate entry? What if there are fifty potential entrants? Mike Shor 24

Survivor Immunity Challenge There are 21 flags Players alternate removing 1, 2, or 3 flags The player to take the last flag wins Mike Shor 25

Unraveling Mike Shor , 97 take gro w 98, , , , , 1 take grow 2, 69, 34, 12  1234

Unraveling Equilibrium: take, take, take, take, take, … Remember:  An equilibrium specifies an action at every decision node  Even those that will not be reached in equilibrium Mike Shor 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? Mike Shor 28

The Game Mike Shor 29 E1 out in M fight acc E2 out in fight acc M E3

Looking Forward … In the last period: No reason to fight final entrant, thus ( In, Accommodate ) Mike Shor 30 E M $0, $100 + previous –50, –50 + previous 50, 50 + previous out in acc fight

… 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! Mike Shor 31

Breakfast Cereals A small sampling of the Kellogg’s portfolio Mike Shor 32

Breakfast Cereals Mike Shor 33 product development costs: $1.2M per product less sweet more sweet sales (in thousands)

Breakfast Cereals Mike Shor 34

Breakfast Cereals Mike Shor less sweet more sweet sales (in thousands)

First Product Entry Mike Shor 36 Profit = ½ 5(600) – 1200 = less sweet more sweet SCENARIO 1 sales (in thousands)

Second Product Entry Mike Shor less sweet more sweet Profit = 2 x 300 = 600 SCENARIO 2 sales (in thousands)

Third Product Entry Mike Shor less sweet more sweet Profit = 300 x 3 – 240 x 2 = 420 SCENARIO 3 sales (in thousands)

Competitor Enters Mike Shor less sweet more sweet Profit = 300 x = 360 SCENARIO 4 sales (in thousands)

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 Mike Shor 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. Mike Shor 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 in a three-way race, then A beats B in a two way race The only political procedure that always guarantees the above is a dictator  No voting system avoids strategic voting Mike Shor 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 Mike Shor 43