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Applied Game Theory Lecture 6

Applied Game Theory Lecture 6. Pietro Michiardi. Recap from last time. A Market Game (1). Assume there are two players An incumbent monopolist ( MicroSoft , MS) of O.S. A young start-up company (SU) with a new O.S.

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Applied Game Theory Lecture 6

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  1. Applied Game TheoryLecture 6 Pietro Michiardi

  2. Recap from last time

  3. A Market Game (1) • Assume there are two players • An incumbent monopolist (MicroSoft, MS) of O.S. • A young start-up company (SU) with a new O.S. • The strategies available to SU are:Enter the market (IN) or stay out (OUT) • The strategies available to MS are:Lower prices and do marketing (FIGHT) or stay put (NOT FIGHT)

  4. A Market Game (2) • What should you do? • Analyze the game with BI • Analyze the normal form equivalent and find NE (-1,0) MS F IN SU NF (1,1) OUT (0,3)

  5. A Market Game (3) NF F (-1,0) MS F IN SU IN NF (1,1) OUT OUT (0,3) Nash Equilibrium (IN, NF) (OUT, F) Backward Induction (IN, NF) This is a NE, but relies on an incredible threat

  6. Informal discussion on how to build up your “mojo” Reputation

  7. A more elaborate setting (1) • Suppose there is one firm • The firm holds a monopoly in ten different markets • In each market, there are reasons to believe the firm will face an entrant • Assume each entrant will come in order

  8. A more elaborate setting (2) • Let’s try to see (intuition first) what happens when each entrant, in order, decides whether to step in the market or stay out • Question: for each entrant, is the monopolist going to fight? • Let’s play, I will be the monopolist

  9. A more elaborate setting (3) • Last time, we analyzed the game as a single market and the outcome (using backward induction) was that as the entrant decided to step in, the monopolist should not fight • This time, instead, there are reasons to believe the monopolist will try to establish a reputation as being a tough firm

  10. A more elaborate setting (4) • The intuition tells us that early fights may keep later entrants out of the markets • Question: what’s worrying about this argument on establishing a reputation? • How should we analyze such a sequential game?

  11. A more elaborate setting (5) • Let’s use backward induction and start from the last entrant • What happened in the game we played? I didn’t fight the last entrant. But why? • When we look at ten markets, the game seems very complicated • But if we look at the last entrant only, the game is nothing but the single market game!

  12. A more elaborate setting (6) • In the last market, we know what the monopolist should do: do not fight, and the entrant should step in! • There are no incentives to establish a reputation for subsequent markets, as there aren’t any • Question: what happens in the 9th market then?

  13. A more elaborate setting (7) • Since we are in a setting of perfect information, the 9th entrant knows where she stands • As the 9th entrant knows that in the last round, the monopolist is not going to fight, because there’s no point • Whatever the 9th entrant will do, the monopolist will let the 10th entrant in • There is no point for the monopolist to establish a reputation of fighting at the 9th market  Hence, the 9th entrant will step in

  14. A more elaborate setting (8) • But now we know what’s going to happen when we look at the 8th market • The 8th entrant knows that whatever she will do, in any case the monopolist will let the 9th and 10th entrants in  The 8th entrant will step in, and so on…

  15. A more elaborate setting (9) • Using backward induction we arrived at a completely different result than what we actually played in class • Nevertheless, the idea of establishing a reputation sounded right! • Next, let’s try to discuss more about the concept of reputation

  16. Reputation (1) • To make our intuition work, let’s try to introduce a new idea • Assume there’s a small chance (say 1%), that the monopolist firm is “crazy” • This implies that the payoffs are not exactly the same as in the single market game we looked at • It’s like the monopolist, sometimes, actually prefers to fight just for the fun of it

  17. Reputation (2) • Now, let’s look at the 1st entrant: she knows there’s a 1% possibility that the monopolist will go bonkers and fight • If there was only one market, we would be done: the entrant should step in • When there are 10 markets, things are different. Let’s see why…

  18. Reputation (3) • Assume that the 1st entrant thinks that with 0.9 probability the monopolist will not fight and she enters • What if the monopolist goes crazy and fight? What happens to the other 9 entrants?  Subsequent entrants would modify their beliefs and assume a higher probability for the monopolist to go bonkers

  19. Reputation (4) • Entrants would start believe that the monopolist is indeed crazy and step out of the markets!  The small possibility that the monopolist would be crazy allowed to build a reputation that keeps entrants out of the markets

  20. Reputation (5) • Now, this argument can be strengthened • Assume that, in fact, the monopolist is not crazy • By acting as if she was crazy early on in the game, she was able to scare the entrants • In game theory, we always assume players know how to put themselves in others’ shoes  Hence, the entrants should know that there’s a possibility that the monopolist is not crazy but she’s acting out

  21. Reputation (6) • Entrants stay out not only because they think the monopolist is crazy • They stay out also because they think that even if the monopolist was not crazy, she would fight in any case! • We just saw that irrespectively of the monopolist being crazy or not, she will act like crazy. Hence, a rational entrant would know that and would not update her beliefs  entrants learn nothing by observation • Question: Can this be an equilibrium?

  22. Reputation (7) • To answer the question, let’s use backward induction and go to the 10th market • The 10th entrant, learns nothing about the monopolist being crazy or not • As such, she would still believe the monopolist could be sane with .99 probability  The 10th entrant would step in, and this would unravel back to the 1st market The answer to the question involves mixed strategies: a monopolist should (with some probability) act like crazy, build a reputation and keep entrants out of the markets

  23. Chain Store Paradox • What we just discussed informally has a name: it’s called the chain store paradox (by Selten, Nobel prize for this contribution) • Reputation is a key concept: by introducing a small probability to play “crazy” helps in many cases • “Short fuse” people, Markets, Doctors, Accountants, Hostage negotiations, …

  24. Or when is more important than what Duels

  25. The Duel Game (1) • Two players, with a gun loaded with one bullet • They stand face to face at a certain distance and the strategies available are: • SHOOT • GET ONE STEP CLOSER • As the distance between the two players is large, there’s a possibility that the shooter will miss • In that case, there’s no second chance

  26. The Duel Game (2) • There are many examples in which duels arise • Historical • Sports: e.g. Tour de France • Economics: R&D efforts to come out with a new product • Duel games have a unique feature we did not encounter yet • The strategic decision is not about what to do, butabout when

  27. The Duel Game (3) • Let’s introduce some notation • Let Pi(d) be player i’s probability of hitting if i shoots at distance d • Example:

  28. The Duel Game (4) • Assumptions: • At distance d=0 the probability of hitting the opponent is 1 • As the distance increases, the probability of hitting the opponent decreases • The two players have different abilities • The two abilities are known by the two players • In the graph shown before, who’s the better shot?

  29. The Duel Game (5) • Question: what do you think it is going to happen, given the game we outlined? • We know that player 1 is better at shooting than player 2 • Is player 1 going to shoot first? • Is player 2 going to shoot first?

  30. The Duel Game (6) • Some possible arguments: • Player 1 should shoot first because in the end he’s better at shooting • Player 2 should shoot first because he knows that player 1 is the best shot and he’s willing to take the chance to shoot before being shot • This line of reasoning is called preemption • It uses concepts of dominance and backward induction

  31. The Duel Game (7) • The analysis of this game is not obvious • We want to answer the following questions: • Who’s going to shoot first? • At exactly what distance? • Let’s start with some facts

  32. The Duel Game (8) • Assume we start with player 1 • Assume player 1 believes player 2 is not going to shoot in the next move • Question: what should player 1 do then?

  33. The Duel Game (9) FACT (A) • Assuming no-one has shot yet, if player i knows, at distance d, that player j will not shoot at distance d-1, then player i should not shoot (at distance d)

  34. The Duel Game (10) • Assume we start with player 1 • Assume player 1 believes player 2 is going to shoot in the next move • Question: what should player 1 do then?

  35. The Duel Game (11) FACT (B) • Assuming no-one has shot yet, if player i knows, at distance d, that player jwill shoot at distance d-1, then player i should shoot if player i’sprobability of hitting at distance d is larger or equal than player j’sprobability of missing at distance d-1

  36. The Duel Game (12) • Formally, player i should shoot at distance d (if she believes player j will shoot at distance d-1) if and only if:

  37. The Duel Game (13)

  38. The Duel Game (14) • Claim: the first shot should occur at distance d* • No one should shoot before d*, by dominance • At d* there is no dominance, we need to use backward induction: we need to know what are the beliefs of what the opponent will do in the next move

  39. The Duel Game (15) • Let’s start at d=0 and assume player 2 is choosing • Player 2 should shoot (prob. 1 of winning) • Now, we are at d=1 and player 1 is choosing • If then player 1 should shoot • Now, we are at d=2 and player 2 is choosing • If then player 1 should shoot

  40. The Duel Game (16) • So the answer to our question is: who shoots first is not necessarily the better or the worse shooter, but whoever’s turn it is first at d* • d* is determined by the joint abilities of the players

  41. Repeated games, discounted payoffs and a little bit of algebra Bargaining games

  42. Ultimatum game (1) • There are two players, player 1 and player 2 • Player 1 is going to make a “take it or leave it” offer to player 2 • This offer concerns a “pie”, that’s worth $1 • Player 1 is given a pie and has to decide how to divide it (hence the value each gets) • (S, 1-S) • E.g. (0.75$, .25$)

  43. Ultimatum game (2) • Player 2 has two choices: • Accept the offer • Decline the offer • If player 2 accepts: • Player 1 gets S, player 2 gets 1-S • If player 2 declines: • Player 1 and player 2 get nothing

  44. Ultimatum game (3) • First thing to notice is that this game doesn’t look like the players are really bargaining • Let’s try to play this game in class to get some intuitions about it • I provide the $1 pie

  45. Ultimatum game (4) • It turns out that in this game a lot of people would reject offers • Let’s try to analyze it using backward induction • Let’s start with the receiver of the offer, choosing to accept or refuse (1-S) • Assuming player 2 is trying to maximize her profit, what should she do?

  46. Ultimatum game (5) • Player 2 is choosing between 0 and 1-S • She should always accept the offer! • So why there were so many offers that were rejected? • Why it seems that the games “converges” to an even split, even if this is not what backward induction predicts?

  47. Ultimatum game (6) • Backward induction is giving a clear prediction: • Player 2 should always accept the offer • Player 1 should offer essentially nothing • Question: why there seem to be a mismatch between BI and reality?

  48. Ultimatum game (7) • Reasons why player 2 may reject: • Pride • She may be sensitive to how her payoffs relates to others • Indignation • Player 2 may want to “teach” a lesson to Player 1 to offer more

  49. Ultimatum game (8) • What we really played is a one-shot game • If we have played it more than once, it makes sense to revisit the concept of reputation • By rejecting an offer, player 2 would also induce player 1 to obtain nothing, which may be an incentive for player 1 to offer more in the next round of the game • Question: Why is a fair (50,50) share so focal here?

  50. Ultimatum game (9) • Lesson learned: even in very simple games, we should be careful about the results backward induction provides, especially if we study real world problems

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