Applied Game Theory Lecture 3

1. Applied Game TheoryLecture 3 Pietro Michiardi

2. Recap (1) • We started to study imperfect competition • We used the Cournot Model • What happens between monopoly and perfect competition? • We used three approaches to answer • Theoretic approach + calculus • Theoretic approach + graphical representation • Economics insights

3. Recap (2) • This is the general approach you use when modeling a problem with Game Theory • You formally solve the problem, with handy mathematical tools • You graphically solve the problem to gain insights and intuition to the problem • You “translate” your findings in the “real world” and figure out if they make sense

4. Recap (3) • In Cournot Equilibrium we have: • Saddle points sit in between monopoly and perfect competition Quantity produced is less than perfect competition but more than monopoly Industry profit is less than monopoly but larger than perfect competition • In our model we set quantities and we let prices take care of themselves and settle

5. Bertrand Competition • What if we take a somehow more realistic view and let Firms decide on prices instead of quantities? • Companies compete on prices and let quantities settle as a consequence of imperfect competition • Let’s define the game

6. Bertrand Model (1) • Players: 2 Companies, E.g.: Coke and Pepsi • Producing perfect substitutes • Constant marginal costs • Strategies: Companies set prices, p1, p2 • Let strategy si be: 0 ≤ pi ≤ 1 • Firms maximize their profits

7. Bertrand Model (2) • Where do quantities come from? where p is the lowest of the prices (p1, p2)  The demand for company 1 would be:

8. Bertrand Model (3) • What are the payoffs? Revenues Costs

9. Bertrand Model (4) • Observations • This is the same basic model we used in Cournot • Before: Firms set quantities, Market determine prices • Now: Firms set prices, Market determine quantities • How to find a NE of this game? • NOTE: calculus is not going to help here • Why?

10. Bertrand Model (5) Best response for Firm 1 • Assume p2 < c • What is the BR in this case?  get out of the market! • Why? • How?

11. Bertrand Model (6) Best response for Firm 1 • Assume p2 > c • What is the BR in this case?  Undercut Firm 2! • Why? • How? • What’s the corollary condition?

12. Bertrand Model (7) Best response for Firm 1 • Assume p2 > pMON • What is the BR in this case?  Be a monopolist! • Why? • How?

13. Bertrand Model (8) Best response for Firm 1 • Assume p2 = c • What is the BR in this case?  Price at least as much as Firm 2 • Why? • How?

14. Bertrand Model (9) • Summary for BR1(p2)

15. Bertrand Model (10) • So now we know the BR for both Firms (symmetric game), what is the NE of the game? • The NE is for both companies to set their prices exactly equal to the marginal costs! • Let’s check this is a NE

16. Bertrand Model (11) • Suppose we have (p1, p2) such as: • What would Firm 1 do? • Now what would Firm 2 do? • … you can imagine we’re we heading

17. Bertrand Model (12) • Hence, the NE = (c,c) • The profit for both Firms is zero • The outcome is perfect competition • Same setting as Cournot, we only changed strategy set, we got a completely different outcome!

18. Remarks on Bertrand Model • We hardly believe this to be representative of reality • We need to relax some assumptions here • We would like to get back at imperfect competition • How can we do that?

19. Linear City Model (1) • NOTE: do this at home, it’s a good exercise for the exam • The assumption we change on the Bertrand Model is that products are not identical anymore • And this is somehow more realistic • Despite being priced equally, you can discern from a beer to another

20. Linear City Model (2) • Players: 2 Firms, E.g.: Coke and Pepsi • Constant marginal costs • Strategies: Companies set prices, p1, p2 • Let strategy si be: 0 ≤ pi ≤ 1 • Firms maximize their profits

21. Linear City Model (3) • Each consumer chooses the product whose total cost to her is smaller • Similar to the demand stated before, smaller price wins • We include a transport cost • Example: consumer at position y pays Firm 1 Firm 2 0 y 1

22. Linear City Model (3) • Assumptions: • Uniform distribution of consumers across the city • Consumers buy only one product from Firm 1 or Firm 2 • Generalization: • The linear city model can be used to think about a “dimension” of the product (think of beer, Bud light vs. Guinness)

23. Let’s move from economics to politics

24. The Candidate-Voter Model (1) • This is an extension to the model we saw during the first lecture, i.e., the Downs and Hotelling Model • Basically we have the same setting, but the game is a little bit different

25. The Candidate-Voter Model (2) • We assume even distribution of voters • Voters vote for the closest candidate • New assumptions: • The number of candidates is not fixed (Endogenous) • Candidates cannot choose their positions • Each voter in the model can be a potential candidate Left wing Right Wing

26. The Candidate-Voter Model (3) • Let’s now describe the game • Players: voters/candidates • Strategies: run for the election or don’t • Voters vote for closest running candidate • Candidate wins with plurality (flip if ties) • Payoffs: • Prize if win = B • Cost for running = C, with B = 2 C • Disutility (cost) if winning candidate is far from your political position = |x-y|

27. The Candidate-Voter Model (4) • Example: • If Mr. X enters and wins  payoff = B – C • If Mr. X enters but Mr. Y wins  payoff = -C - |x-y| • If Mr. X stays out  payoff = - |x-y| • Let’s put some real numbers: • Suppose we have N = 17 players • B = $2000 • C = $1000 • Each place is worth a 1/17th of $1000  for each position away form you the winner is, you loose ~ $60

28. The Candidate-Voter Model (5) Y X • Who is going to win the election? • Is this a Nash Equilibrium? • Y runs payoff = - $1000 - $240 • Y stays put  payoff = -$240 • A player can deviate by staying put L R

29. The Candidate-Voter Model (6) Y X • Who is going to win the election? • Is this a Nash Equilibrium? • Y runs payoff = - $1000 - $60 • Y stays put  payoff = -$60 • A player can deviate by staying put L R

30. The Candidate-Voter Model (7) X • Who is going to win the election? • Is this a Nash Equilibrium? • What if another candidate enters? Deviation: run L R X Y L R

31. The Candidate-Voter Model (8) • Looks like we understood the mechanics Question: Is there a NE with zero candidates running? Question: given an odd number of candidates Is there a NE with only 1 candidate running?

32. The Candidate-Voter Model (9) • Is this an equilibrium? • What happens if someone else runs? • Who is the winner in case orange player runs? L R L R

33. The Candidate-Voter Model (10) Y X • Is there an NE with 2 candidates? • What about the example above? • We need to check for all possible deviations L R

34. The Candidate-Voter Model (11) Deviation 1: Z enters the scene  not only Z looses, but it pushes the winner further away Z L R Deviation 2: Z enters the scene  Z simply looses Z L R Deviation 3: X drops  X runs, then expected payoff is 50% B-C , 50% -C - |x-y| X drops, then payoff is –C - |x-y| for sure!! L R

35. The Candidate-Voter Model (12) • Summarizing: • NE with 0 candidates  NO • NE with 1 candidate  Yes, if odd # of voters and centrist candidate • NE with 2 candidates  YES, if equally distant from the center! • Can we elaborate more on this last point?

36. The Candidate-Voter Model (13) • Let’s recap first • There are many NE, not all “at the center” • Entry can lead to a more distant candidate winning X Y L R

37. The Candidate-Voter Model (14) • If runners are too far apart, there’s an incentive for a third party to run and win • Question: how far apart can two equilibrium candidates be? X Y L R

38. The Candidate-Voter Model (15) ?? • If runners are exactly at 1/6 and 5/6 a middle runner could win with probability 1/3 • If they slightly converge towards the center, the middle candidate is squeezed out • Although there is not a full thrust to aggregate exactly at the center (cf. Downsian model) there is still a force pushing candidates towards the center 1 2/6 1/2 4/6 5/6 0 1/6

39. Game theory lesson • Basically we’ve seen that the “guess and check” technique is very effective • HINTS: • Be systematic when guessing • Be careful when checking: do not ignore hidden deviations of players

40. Let’s now move from politics to sociology

41. The Location Model (1) • Assume we have 2N players in this game • Players have two types: tall and short • There are N tall players and N short players • Players are people that need to decide in which town to live • There are two towns: East town and West town • Each town can host no more than N players

42. The Location Model (2) • Players: 2N people • Strategies: East or West town • Let’s put some numbers • We have 140 short people and 140 tall people • Each town can host 140 people • As usual, we’re missing something: payoffs

43. The Location Model (3) Utility for player i 1 1/2 0 # of your type in your town 70 140

44. The Location Model (4) • The idea is: • If you are a minority in your town you get a payoff of zero • If you are in majority in your town you get a payoff of ½ • If you are well integrated you get a payoff of 1 • People would like to live in mixed towns, but if they cannot, then they prefer to live in the majority town

45. The Location Model (5) • Let’s put a few more rules to define the game • We assume a simultaneous move game • Unrealistic, but will do for now • We assume that if the number of people choosing a particular town is larger than the town capacity, the surplus will be redistributed randomly

46. Tall player West Town Short player East Town • This is the initial picture • We assume an initial choice for all players • What we are going to do is to simulate players’ action in repeating the game

47. Tall player West Town Short player East Town • First iteration • For tall players • There’s a minority of east town “giants” to begin with • For short players • There’s a minority of west town “dwarfs” to begin with •  short players will want to switch towns (WE) •  tall players will want to switch towns (EW)

48. Tall player West Town Short player East Town • Second iteration • We keep the same trend •  short players will want to switch towns (WE) •  tall players will want to switch towns (EW) • Few exceptions that still didn’t understand the game • What is their payoff?

49. Tall player West Town Short player East Town • Third iteration • What happened? • People got segregated

50. The Location Model (6) • The players ended up being segregated • However, the payoff curve didn’t say so: • People would have preferred to be in an integrated town • What happened? Why did we end up like this?