Dynamic Games

1. Dynamic Games

2. Overview • In this unit we study: • Combinations of sequential and simultaneous games • Solutions to these types of games • Repeated games • How to use dynamics to build self-sustaining agreements

3. Sequential and Simultaneous Games • There are many situations where the strategic situation has both simultaneous and sequential elements • Examples: • Decision by a firm to enter a market followed by competition in pricing and advertising • Decisions by candidates to run for office followed by voting • Attempts at legal settlements followed by trial in the event no settlement is reached

4. How to Analyze these Games? • In sequential games, we saw that it paid to look forward and reason back • Find the best decision a player can make on reaching a point in the game • “Prune” the game tree to eliminate worse (dominated) decisions • In simultaneous games, we looked for a best response to a best response (Nash equilibrium). • Project the strategy of a rival • Choose a best response to that strategy • Check if the rival would want to change his “projected” strategy in view of your move.

5. New Elements to Dynamic Games • History matters • Strategy is now based not only on projections of the future and the present but also on the past. • The process by which you arrived at a point in the game might matter • Alternative futures matter • Histories of the game that no one contemplated as arising can play a key role in influencing outcomes

6. Entry • Consider the following situation: • An incumbent firm is presently operating in some profitable market • A potential rival firm is considering entering this market • Upon entry, the rival and the incumbent simultaneously decide whether to fight or not fight • If they both fight, then the profitability of the market is such that the rival would be better off staying out • If they do not fight, the rival would be better off entering

7. Game Tree in rival out Incumbent is row player. Payoffs are (incumbent, rival) 2a, e

8. Description • One interpretation of this diagram is the following: • There are three levels of profitability in the market: • High (when no one fights) • Medium (when 1 firm fights) • Low (when both fight) • The product of fighting is to capture market share from the rival • If both fight or both don’t fight, market share is 50-50 • If one fights and the other doesn’t the fighter gains market share at the expense of the rival

9. Goal • Your goal is to provide an analysis of under what conditions to enter this market. • What are the key things to think about in making this decision?

10. Example 1: Large Market Share Capture • Suppose that the profitability of the market is: • Big (No one fights)= 32 • Medium (One firm fights)= 25 • Small (Both fight) = 16 • The outside option of a rival who does not enter is 11 • When 1 firm fights and the other does not, the fighter obtains 80% market share

11. Game Tree: Example 1 in rival out 32, 11

12. Analysis • If the rival enters: • Incumbent’s best response to don’t fight is to fight • Incumbent’s best response to fight is to fight • The situation is symmetric for the rival • Therefore: • If enter, then (fight, fight)

13. Game Tree: Example 1 in rival out Since the rival anticipates a fight on entering, it is better not to enter 32, 11

14. Game Tree: Example 1- Generalized in rival out • a > e > b (It only pays to enter absent a fight) • c > a • b > d 2a, e

15. Game Tree: Example 1- Generalized • a > e > b (It only pays to enter absent a fight) • c > a • b > d in rival out Notice that the game after entry is a Prisoner’s dilemma In this case the incumbent uses the b, b outcome to successfully deter entry 2a, e

16. Comments • Notice that what didn’t happen –entry – had a profound effect on what did • The rival could count on the fact that entry combined with the temptation to grab market share would lead to a fight • Therefore, it paid to stay out of this market.

17. Example 2: Smaller Market Share Grab • Suppose that the market sizes are again • Big = 32 • Medium = 25 • Small = 16 • The outside option of a rival who does not enter is still 11 • When 1 firm fights and the other does not, the fighter obtains 60% market share • Now what happens?

18. Game Tree: Example 2 in rival out 32, 11

19. Analysis • If rival enters: • Incumbent’s best response to don’t fight is don’t fight • Incumbent’s best response to fight is don’t fight • Same for rival • So neither fight if rival enters • If rival does not enter • Incumbent is free to threaten to do whatever it likes • In particular, it can threaten to fight • In which case it pays for the rival to stay out

20. Equilibria • Rival enters, neither firm fights • Rival doesn’t enter, incumbent threatens to fight if it did enter • Notice that now entry deterrence depends crucially on the rival’s beliefs about the incumbent’s response • If the rival is convinced that the incumbent will be aggressive, it should not enter • Since the rival chooses not to enter, choosing to actually be aggressive is a best response by incumbent

21. Game Tree: Example 2 Generalized a > e > b in rival out 1. a > c 2. d > b 2a, e

22. Game Tree: Example 2 Generalized a > e > b 1. a > c 2. d > b in rival out Even though it is a dominant strategy for incumbent to not fight It can deter entry by threatening. Since in the even of successful deterrence, the threat is no tested this is still a best response for the incumbent 2a, e

23. Comments • So it would seem that if the incumbent can affect the rival’s beliefs, it is possible to deter entry even in this framework.

24. Choosing Equilibria • The prospect that a threat which is costly to carry out might succeed in a situation like this posed a problem for game theory • Is there some rational means to choose between the equilibria?

25. Subgame Perfect Equilibria • We’ll generalize the idea of look forward, reason back the following way: • Rationality Axiom 2: When presented with any history of the game (even an unexpected one), players should choose best responses to future beliefs • Formally, we require that players choose optimizing strategies everywhere in the game

26. Example 2: Refined • Recall that not fighting was a dominant strategy for each of the players if the rival enters • Therefore, despite incumbent’s threats to the contrary • Rival should anticipate that its entry will not lead to fighting • Therefore, it pays to enter. • Entry deterrence is not credible in this case.

27. Example 3: Shrinking Markets • Now consider a variation of example 2. • Suppose that when either firm fights, it still gains 60% market share but the profitability of the market shrinks to a greater extent than before. • Does this change rival’s view of the incumbent’s threat to fight?

28. Example 3: Specifics • Suppose that the market sizes are now • Big = 32 • Medium = 18 • Small = 16 • The outside option of a rival who does not enter is 11 • When 1 firm fights and the other does not, the fighter still obtains 60% market share

29. Game Tree: Example 3 in rival out 32, 11

30. Analysis • The best response to not fighting is not to fight • The best response to fighting is to fight • Therefore, there are 2 equilibria following entry • If rival anticipates a fight, it should not enter • If it anticipates no fighting, it should • Hence there are 2 equilibria of the dynamic game: • Out -> Fight, Fight • In -> Don’t fight, Don’t fight

31. Comments • Notice that making competition more disruptive on entry actually improves the credibility of the threat by the incumbent • In many situations it is possible to control how destructive competition will be in markets • The subtlety here is that it’s in the incumbent’s interest to make the destructiveness of competition more rather than less

32. Game Tree: Example 3 Generalized a > e > b in rival out 1. a > c 2. b > d 2a, e

33. Game Tree: Example 2 Generalized a > e > b 1. a > c 2. b > d in rival out Now this is a coordination game if the rival enters. Either both firms can coordinate on not fighting, or they can coordinate on fighting Either is self-sustaining and the fighting outcome deters entry. 2a, e

34. Key Conclusions • The idea of subgame perfection is to assess the credibility of threats. • We’ll return to this issue in the next class • Threats which are not self-sustaining if carried out, should correctly be viewed with skepticism • To make a threat credible, it can sometimes serve the interest of the incumbent to destroy profitability of the market in the event of entry.

35. Repeated Games • We turn now to repeated games • These are games where players are involved in the same (or similar) strategic situation for many periods in a row. • The key insight here will be that we can use the future to affect the outcome in the present.

36. Example 1 Revisited • Suppose that entry has occurred and that the situation is as in example 1

37. Example 1: Game Table • Suppose • a > b • Competition is destructive to profitability • c > a • Market share grabs are profitable • b > d • Fighting back is better than being a victim of a grab

38. Analysis • We determined that fighting was the inevitable outcome • With consequent decrease in the profitability of the market • Suppose that the firms will compete for 2 periods instead of 1? • Firm 1 and 2 agree to the following: • Don’t fight in either period • If either of us fights in the first period, fight in the second • Will this work?

39. Carrots and Sticks • Each firm is holding out a carrot—the promise of a in both periods • And a stick, the threat of b in the second period • To try to deter the temptation to grab market share (and get c) • This could work if 2a > c + b

40. Look Forward Reason Back • But there’s a problem here: • In period 2, there’s no stick and no carrot • So each firm will be tempted to fight and succumb to that temptation • And this “reverberates” back to period 1 • Each firm knows that there is no “carrot” in the second period---only the “stick” • So there’s nothing to deter the temptation to fight in period 1 • Hence the firms will fight in both periods

41. Generalizing • The principle applies for any set number of periods. • Since there’s nothing to promote good behavior at the end of the game, firms will fight then • And this reverberates backward throughout the game • The conclusion is a sad one: • If the game has any set ending time, the firms will fight in every period

42. Infinite Repetition • Suppose there is no fixed endpoint to the game • Instead the firms expect the game to be infinitely repeated • This is an abstraction---think of it as a game being repeated for a really long time with no one knowing exactly when it will end • Now can the firms cooperate?

43. Tit-for-Tat Strategies • Suppose the firms make the following deal: • We agree not to fight • If my rival fights, I’ll fight in the next period • Then the war is over and we’ll resume not fighting • Will this work? • Now in every period the firm must weigh the gains from cheating (i.e. the temptation) • c – a • Versus the cost in the future (the carrot and stick) • a – b • If the cost exceeds the temptation, each firm will refrain from fighting. • a – b > c – a

44. Variations • Suppose the carrot and stick in the above agreement are not enough to overcome the temptation • i.e., a – b < c – a • Is all lost? • No. Because there is always a future, the size of the “stick” can be increased

45. Tit-for-2 Tats • Suppose the firms make the following deal: • We agree not to fight • If my rival fights, I’ll fight for the next 2 periods • Then the war is over and we’ll resume not fighting • Will this work? • Now in every period the firm must weigh the gains from cheating (i.e. the temptation) • c – a • Versus the cost in the future (the carrot and stick) • 2 x (a – b) • If the cost exceeds the temptation, each firm will refrain from fighting. • 2 x (a – b) > c – a

46. Key Point • The larger is the temptation or the weaker the punishment available in any one period • The longer the threat of “war” needs to be to deter cheating. • The size of the stick needs to be calibrated to the upside from cheating. • Obviously the promise of infinite war leads to the largest possible “stick”

47. Discounting • All of this assumed that profits in the future were worth the same as those in the present • Of course, they’re not • Suppose that we discount profits in the future by the real interest rate r. • How does this change the analysis?

48. Tit for n Tats • If we play tit-for-tat: • NPV of temptation = c – a • NPV of the threat = (a – b)/(1 + r) • Notice that as the interest rate increases, the punishment associated with the threat declines

49. Perpetual Punishment • In a world with positive real interest rates, even the threat of perpetual punishment may not be enough to stave off fighting • Recall the perpetuity formula: • What is the NPV of an asset has a cash flow C starting 1 period in the future and lasting in perpetuity? • NPV = C/r

50. Analysis • Suppose we threaten punishment forever after a defection. • NPV of Temptation = c – a • NPV of punishment = (a – b)/r • For higher real interest rates, even the most severe possible punishment loses efficacy • If this threat is insufficient to deter cheating, nothing will deter it.