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# Applied Game Theory Lecture 2

Applied Game Theory Lecture 2. Pietro Michiardi. Recap …. We introduced the idea of best response (BR) do the best you can do, given your belief about what the other players will do We saw a simple game in which we applied the BR idea and worked with plots. Soccer: Penalty Kick Game (1).

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## Applied Game Theory Lecture 2

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1. Applied Game TheoryLecture 2 Pietro Michiardi

2. Recap … • We introduced the idea of best response (BR) • do the best you can do, given your belief about what the other players will do • We saw a simple game in which we applied the BR idea and worked with plots

3. Soccer: Penalty Kick Game (1) goalie r l • Payoffs approximate the probabilities of scoring for the kicker, and the negative of that for the goalie • Assumption: we ignore the “stay put” option for the goalie • Example: • u1(L,l) = 4  40% chance of scoring • u1(L,r) = 9  90% chance of scoring L kicker M R

4. Penalty Kick Game (2) • What would you do here? • Is there any dominated strategy? • If we stopped to the idea of iterative deletion of dominated strategies, we would be stuck! • If you were the kicker, were would you shoot?

5. Expected Payoff E[u1(L, p(r) )] E[u1(R, p(r) )] 10 8 6 4 E[u1(L, p(r) )] 2 0 x 0.5 y 1 p(r) “belief”

6. Penalty Kick Game (3) • What’s the lesson here? • Assume for a moment these numbers are true • If the goalie is jumping to the right with a probability less than 0.5, then you should shoot …. Lesson: Don’t shoot to the middle

7. Lesson 1: Do not choose a strategy that is never a BR to any(*) belief (*) any means all probabilities

8. Penalty Kick Game (4) • Notice how we could eliminate one strategy even though nothing was dominated • With deletion of dominated strategies we got nowhere • With BR, we made some progress… • Can we do better? What are we missing here?

9. Penalty Kick Game (5) • Right footed players find it easier to shoot to their left! • The goalie might stay in the middle • The probabilities we used before are artificial, what about reality? • What about considering also the speed? • And the precision?

10. Expected Payoff 10 8 6 4 2 0 0.5 1 p(r) “belief” See what happens? If you are less precise but strong you’d be better off by shooting to the middle

11. The Partnership Game (1) • Two individuals (players) who are going to supply an input to a joint project • The two individuals share 50% of the profit • The two individuals supply efforts individually • Each player chooses the effort level to put into the project (e.g. working hours)

12. The Partnership Game (2) • Let’s be more formal, and normalize the effort in hours a player chooses • Si = [0,4] • Note: this is a continuous set of strategies

13. The Partnership Game (3) • Let’s now define the profit to the partnership Profit = 4 [s1 + s2 + b s1 s2] • Where: • si = the effort level chosen by player i • b = synergy / complementarity • 0 ≤ b ≤ ¼ • Why is there the term s1 s2 ?

14. The Partnership Game (4) • What’s missing? Payoffs! u1(s1 , s2) = ½ [4 (s1 + s2 + b s1 s2)] - s12 u2(s1 , s2) = ½ [4 (s1 + s2 + b s1 s2)] - s22 • That is: • Players share the profit in half • They bear a cost proportional to the square of their effort level • Note: payoff = benefit - cost

15. The Partnership Game (5) • Alright, how can we proceed now? • Let’s analyze this game with the idea of BR • But how can we draw a graph with a continuous set of strategies? • Recall the definition of best response

16. The Partnership Game (6) • We are going to use some calculus here ŝ1 = arg max { 2 (s1 + s2 + b s1 s2) - s12 }

17. The Partnership Game (7) • So we differentiate: • F.o.c. : 2 (1 + b s2) - 2s1 = 0 • S.o.c. : -2 < 0 ŝ1 = 1 + b s2 = BR1(s2) ŝ2 = 1 + b s1 = BR2(s1)  due to symmetry of the game

18. The Partnership Game (8) • Alright, we have the expressions that tell me: • player i best response, given what player j is doing • Now, let’s draw the two functions we found and have a look at what we can say • Let’s also fix the only parameter of the game: b = ¼

19. s2 5 BR1(s2) 4 3 BR2(s1) 2 1 0 s1 1 3 2 4 5

20. s2 5 BR1(s2) 4 3 2 1 0 s1 1 3 2 4 5

21. s2 5 4 3 BR2(s1) 2 1 0 s1 1 3 2 4 5

22. s2 5 BR1(s2) 4 3 BR2(s1) 2 1 0 s1 1 3 2 4 5

23. s2 BR1(s2) 2 7/4 BR2(s1) 6/4 5/4 1 s1 5/4 7/4 6/4 2

24. The Partnership Game (9) • We started with a game • We found what player 1 BR was for every possible choice of player 2 • We did the same for player 2 • We eliminated all strategies that were never a BR • We looked at the ones that were left, and eliminated those that were never a best response • … • Where are we going to?

25. The Partnership Game (10) s*1 = 1 + b s2 s*2 = 1 + b s1 The intersection  s*1 = s*2  s*1 = 1/(1-b)

26. The Partnership Game (11) • We came up with a prediction on the effort levels • Question: is the amount of work we found previously a good amount of work? • Compared to what? • Question: are the players working more or less than an efficient level?

27. The Partnership Game (12) • Why is it that in a joint project we tend to get inefficiently little effort when we figure out what’s the best response in the game? • NOTE: this is not a PD situation • “I’m gonna let the other work and I’ll shirk” • Why?

28. The Partnership Game (13) • The problem is not really the amount of work • Also, the problem is not about synergy, i.e. the factor b • The problem is that at the margin, I bear the cost for the extra unit of effort I contribute, but I’m only reaping half of the induced profits, because of profit sharing • This is known as an “externaility” • When I’m figuring out the effort I have to put I don’t take into account that other half of profit that goes to my partner • In other words, my effort benefits my partner, not just me

29. The Partnership Game (14) • By the way, how would the situation change by varying the only parameter of the game? • Informally, what we have done so far is to determine the Nash Equilibrium of the game

30. Introducing NE • So in the partnership game we’ve seen what a NE is… • Recall the numbers game: what was the NE there? • Did you play a NE? • Although NE is a central idea in game theory, be aware that it is not always going to be played • By repeating the numbers game, however, we’ve seen that we were converging to the NE

31. Why is it an important concept? • It’s in textbooks • It’s used in many applications • Don’t jump to the conclusion that now we know NE, everything we’ve done so far is irrelevant

32. NE: observations • It is not always the case that players play a NE! • E.g.: in the numbers game, we saw that playing NE is not guaranteed • Rationalitydoes not imply playing NE • What are the motivations for studying NE?

33. NE: motivations (1) NO REGRETS • Holding everyone else’s strategies fixed, no individual has a strict incentive to move away • Having played a game, suppose you played a NE: looking back the answer to the question “Do I regret my actions?” would be“No, given what other players did, I did my best”

34. NE: motivations (2) Self-fullfilling beliefs • If I believe everyone is going to play their parts of a NE, then everyone will in fact play a NE • Why?

35. s2 Let’s play a little bit with this graph: - Graphical way of finding NE 5 BR1(s2) 4 3 2 1 0 s1 1 3 2 4 5

36. Finding NE point(s) • Next we will play some very simple games involving few players and few strategies • Get familiar with finding NE on normal form games • We will have a glimpse on algorithmic ways of finding NE and their complexity

37. A simple game (1) Player 2 c r l • Is there any dominated strategy for player 1/2? • What is the BR for player 1 if player 2 chooses left? • What is the BR if player 2 chooses center? • What about right? • Can you do it for player 2? U Player 1 M D

38. A simple game (2) Player 2 c r l • BR1(l) = M BR2(U) = l • BR1(c) = U BR2(M) = c • BR1(r) = D BR2(D) = r What is the NE? Why? U Player 1 M D

39. A simple game (3) • It looks like each strategy of player 1 is a BR to something • And the same is true for player 2 • Deletion of dominated strategies wouldn’t lead anywhere here… • Would it be rational for player 1 to chose “M”?

40. Another simple game (1) Player 2 c r l • What is the NE for this game? • What’s tricky in this game? • Do BR have to be unique? • Are players happy about playing the NE? U Player 1 M D

41. NE vs. Dominance (1) • We’ve seen how to find NE on a normal form game • We’ve seen how NE relates to the idea of BR • We have a NE when the BR coincide • What is the relation between NE and the notion of dominance?

42. NE vs. Dominance (2) Player 2 • What is this game? • Are there any dominated strategies? • What is the NE for this game? beta alpha alpha Player 1 beta

43. NE vs. Dominance (3) • Claim: no strictly dominated strategies could ever be played in NE • Why? A strictly dominated strategy is never a best response to anything • What about weakly dominated strategies?

44. NE vs. Dominance (4) Player 2 • Are there any dominated strategies? • What is the NE for this game? r l l Player 1 r

45. NE vs. Dominance (5) • First observation: the game has 2 NE! • Informally we’ve seen that a NE can be: • Everyone plays a BR • None has any strict incentive to deviate • What’s annoying here? What is the prediction game theory leads us to? • Is that reasonable?

46. The Investment Game (1) • The players: you • The strategies: each of you chose between investing nothing in a class project (\$0) or invest (\$10) • Payoffs: • If you don’t invest your payoff is \$0 • If you invest you’re going to make a net profit of \$5 (gross profit = \$15; investment \$10)This however requires more than 90% of the class to invest. Otherwise, you loose \$10 • As usual no communication please!!

47. The Investment Game (2) • What did you do? • Who invested? • Who did not invest? • What is the NE in this game?

48. The Investment Game (3) • There are 2 NE in this game • All invest • None invest • Let’s check: • If everyone invests, none would have regrets, and indeed the BR would be to invest • If nobody invests, then the BR would be to not invest

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