1 / 45

Uncertainty and Information

Uncertainty and Information. Incomplete/Asymmetric Information. In games often some players may have an advantage of knowing with greater certainty what has happened or what will happen. (private information)

jorryn
Download Presentation

Uncertainty and Information

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Uncertainty and Information

  2. Incomplete/Asymmetric Information • In games often some players may have an advantage of knowing with greater certainty what has happened or what will happen. (private information) • One may know better his own preferences or payoffs (risk tolerance, patience, peaceful or warlike intentions), or may have better knowledge of his own innate characteristics (skills of an employee, riskiness of an insurance applicant, car condition).

  3. Games with incomplete information • Simultaneous-move →Static Bayesian Game • Sequential-move →Dynamic Bayesian Game -Signaling: the informed player moves first to signal its type (usually costly) -Screening: the less informed moves first to set a mechanism to screen opponents’ types -Cheap-Talk: similar to signaling, but the sender just sends costless talk.

  4. Signaling Game • Games where better informed players (sender) have options to choose actions to implicitly reveal his (true or misleading) type → sequential-move • The better informed player may want to • Conceal information or reveal misleading information • Reveal selected information truthfully

  5. I -16, -10 • Example 1:signaling costs 6 Attacker NI S -6, 0 Defender NS -10, -10 D is Tough (25%) I Attacker 0, 0 NI Nature -11, 5 I Attacker D is Weak (75%) -6, 0 NI S Defender -5, 5 NS I Attacker NI 0, 0

  6. Receiver will use the signal to infer the type of players (using Bayes’ Rule) and then make optimal decisions based on it. • Perfect Bayesian equilibrium (PBE) • Players draw correct inferences (belief) from observation • Players make optimal decisions given inferred information

  7. We ask if those outcomes are equilibria. • Separating Equilibrium (different (types) senders (D) sending different signals) 1. T for S, and W for NS 2. T for NS, and W for S • Pooling Equilibrium 3. Both for S 4. Both for NS • Semiseparating Equilibrium

  8. T for S, and W for NS For the receiver (A), observing S means facing T with certainty, while observing NS means facing W with certainty. (based on the equilibrium specified) • Attacker will choose NI seeing S, and I seeing NS. Next we ask if this equilibrium can sustain, so that no players will deviate.

  9. If T-typed Defender deviates to NS, it will be inferred as W, and be invaded, earning -10. T-typed D will not deviate (compared to -6). • W-typed Defender will not deviate either. (-6 is lower than -5) • An perfect Bayesian equilibrium

  10. T for NS, and W for S For the receiver (A), observing NS means facing T with certainty, while observing S means facing W with certainty. • A will choose NI seeing NS, and I seeing S. • If T-typed Defender deviates to S, will be inferred as W, and invaded, earning -16. T type will not deviate (compared to 0). • However, W-typed Defender will deviate to NS to be inferred as T and not be invaded to earn 0. • This cannot be a PBE

  11. Both for S • The receiver (A) cannot infer additional information, thus using presumed probability (0.25, 0.75) to consider. • Facing S, if the receiver chooses I, it obtains 0.25X(-10)+0.75X5=1.25 • Facing S, if NI, 0.25X(0)+0.75X0=0 • Attacker will Invade seeing S, so that T-typed earns -16, and W-typed earns -11.

  12. However, T-typed Defender is better off choosing NS either A will attack (-10) or not (0). • Similarly W-typed Defender will deviate too. • Not a PBE

  13. Both for NS • Again the receiver (A) cannot infer additional information, thus using prior probability to consider. • Facing NS, if the receiver (A) chooses I, 0.25X(-10)+0.75X5=1.25 • Facing NS, if NI, 0.25X(0)+0.75X0=0 • Attacker will Invade seeing NS so that T-typed earns -10, and W-typed earns -5.

  14. First, W-typed has no incentive to deviate (-11/-6 is lower than -5) • For this to be an equilibrium, A must choose to Invade once seeing S, so that T-typed D will not deviate either. • Note if A always choose NI when it sees S, then T-typed D has an incentive to deviate. However, how will A act depends on its belief. • If A believes with probability p it will be T-typed sender when A incidentally sees S (off-equilibrium path), it will choose I if and only if px(-10)+(1-p)x5>0→p<1/3

  15. However, since W-Typed Defender will always chooses NS, Attacker will always believe it’s T-typed whenever it sees S. Hence its belief (p, the probability it’s T-typed after seeing S) is p=1. • This cannot be a PBE.

  16. Anther way to represent the game -10, -10 -16, -10 I I Tough NS S q p NI NI 0, 0 0.25 -6, 0 Nature Receiver Receiver -11, 5 -5, 5 0.75 I 1-q 1-p I S NS NI NI Weak -6, 0 0, 0

  17. Perfect Bayesian Equilibrium → [(T-type's action, W-type's action), (Receiver’s action seeing S, Receiver’s action seeing NS), Receiver’s belief at 1st info. set (p), Receiver’s belief at 2nd info. set (q)] • A unique pure strategy PBE. • Separating Equilibrium → [(S, NS),(NI, I), p=1, q=0]

  18. Semiseparating Equilibrium An equilibrium where one type of the sender and the receiver mix upon the message from which the later cannot distinguish the former’s type. But the receiver knows the type of the sender when seeing the other message. • In our example, W-typed never chooses S. Hence when A observes S, it must be T-typed. However, when A sees NS, it can come from both types.

  19. Let x be the prob. that T-typed chooses NS. • While the prob. for A to see S is 0.25(1-x). The prob. for A to see NS is the sum of the two T-typed：0.25x W-typed：0.75 • When seeing NS, A’s belief that it’s T-typed will be 0.25x/(0.75+0.25x) • When seeing NS, A’s belief that it’s W-typed will be 0.75/(0.75+0.25x)

  20. NS 75% W-type NS x 25% T-type S 1-x

  21. By the indifference principle, if A mixes I/NI upon seeing NS, he’s indifferent between I：0.25x /(0.75+0.25x)．(-10)+ 0.75 /(0.75+0.25x)．(5) NI：0 • x=1.5>1, indeed I is always preferred to NI from A’s perspective for x<=1. • Observing NS, A always prefer I than NI even it’s likely some are T-type. • Observing A’s optimal decision, T-type will choose S purely. • There’s no semiseparating equilibrium.

  22. I -14, -10 • Signaling Costs are 4 Attacker NI S -4, 0 Defender NS -10, -10 D is Tough (25%) I Attacker 0, 0 NI Nature -9, 5 I Attacker D is Weak (75%) -4, 0 NI S Defender -5, 5 NS I Attacker NI 0, 0

  23. Neither the two separating outcomes, nor the pooling outcome where both types choose S, are PBE. • The only pure strategy PBE is [(NS, NS), (I, I), p<1/3, 0.25]. • There’s also a semiseparating equilibrium where T-type chooses S purely and W-type mixes. (You can verify by yourself that other semiseparating outcomes are not equilibria.)

  24. Given the equilibrium strategy, whenever A observes NS, it must be W-type, and A will Invade. • Given above, A will also mix I and IN observing S. Otherwise W-type will play S or NS purely. (W-type will get certainly -9 if A plays I purely observing S. But W-type can get -5 if he plays NS purely, so that he will not mix. Or it will get -4 if A plays NI purely. Then W-type shall play S purely.

  25. Suppose W-type who mixes sends S with probability x and NS with probability 1-x. • Observing S, the likelihood A is facing T-type is 0.25/(0.25+0.75x), and W-type 0.75x/(0.25+0.75x). • Observing S, if A invades, his expected payoff will be 0.25/(0.25+0.75x)(-10)+ 0.75x/(0.25+0.75x)(5); if A does Not Invade, his expected payoff is 0. A will mix if and only if x=2/3.

  26. Let y be A’s prob. to Invade is if he sees S and 1-y to Not Invade (Note he will play I purely seeing NS). • Again by indifference principle, W-typed will mix if he’s indifferent in S：-9y-4(1-y) NS： -5 • y=0.2, the prob. For A to invade if he sees S.

  27. Homework • Consider a possible scenario between used-car (of Good or Bad qualities) sellers and buyers (who can choose to Buy or Not to Buy (game tree in the next slide). Assume the buyers believe that only 20% of the chance the cars are of good quality. Note in this example the signal (by sending the car fixed first) may benefit the receiver if he buys the car. Find all the PBE.

  28. 5, 2 3, 5 B Good B NS S q p NB NB 0.2 0, 0 -2, 0 Nature Receiver Receiver 0.8 1, 1 3, -3 B 1-q 1-p B S NS NB NB Bad -2, 0 0, 0

  29. Without Signaling • What will happen without the signaling mechanism? • Lemon/Peach (Akerlof, 1970) • In the used cars market, 2/5 is peaches and 3/5 are lemon. A peach is worth $150,000 to a seller and $200,000 to a buyer, and a lemon is worth $50,000 to a seller and $100,000 to a buyer. • Assume there’re many buyers.

  30. Perfect information: two separate markets, with peach for $200,000, and lemon for $100,000. • If neither knows the quality, the car will be sold at $200,000x0.4+$100,000x0.6=$140,000. • Asymmetric information: buyers will only like to pay $140,000, but sellers with good cars will exit the market (similar to Gresham's law) • Adverse Selection.

  31. Screening • Games of incomplete information where the less informed ones setting up a menu of prices to distinguish the opponents of many types. • The job market-hidden information • Principal/agent problem-hidden action

  32. Hidden information • 2 types workers, Able and Challenged • Employers willing to pay $150,000 for type A and $100,000 for type B. • Competitive firms offering wages • Types not observable, but • A screening device-education levels. And a year of education will cost Type A: -$6000 a year of salary Type B: -$9000 a year of salary

  33. How to use the years of education to screen different types? • Suppose any one who has taken n or more years will be considered A type, and anyone who has taken less than n years will be considered B type. • 150,000-6,000n≧100,000 →n≦8.33 • 100,000 ≧150,000-9,000n →n≧5.56

  34. Incentive-compatibility constraints: constraints align the agents’ (applicants’) incentives with the principal’s (employer) desires. • Self-selection, separation of types. • Pooling? (Suppose with prob. p, the worker is type A, 1-p for type C.) • px$150,000+(1-p)x100,000=… • It will be $125,000 if p=50% • The higher p is, the higher the employer would like to pay.

  35. A-typed may not necessarily find the pooling better, but C type will always do. • Even both find it better, the pooling equilibrium may collapse. • If p=0.5, one employer (or a worker of A type) may find it better to offer (to propose) $132,000 for A type with one year of education and will be accepted by the employer. Type C finds it worthless to follow but other A typed will follow… • With smaller portion of A-typed in the pool, the employer will only pay less, and then C-type will find it worth to receive one year of education.

  36. Hidden action • The owner of a firm (the principal) is hiring a manager to undertake a project (worth $600,000 if it succeeds) • 60% of chance it will succeed if the manager’s effort is routine, and 80% if the effort is of high quality. • Suppose a manager of routine effort asks $100,000 and another $50,000 more for extra efforts. This will be accepted. • However, the effort is not observable. • Moral Hazard

  37. Consider a compensation package of a base salary s, and a bonus b if the project succeeds. The manager expects to earn s+0.6b if routine effort s+0.8b if high effort • s+0.8b-(s+0.6b)≧50,000 →b≧250,000 incentive-compatibility constraint

  38. Participation constraint s+0.8b≧150,000 • s*=-50,000 but may be against law → s=0, b=250,000. And the expected total is 200,000. • The owner expect to earn (given the manager will put extra effort 0.8x600,000-200,000=280,000 • Is it worth it for the owner to set up such a menu of price? • If he sticks to the original basic plan, his expected payoff is 0.6x600,000-100,000=260,000 • The owner will go for the incentive mechanism.

  39. Cheap Talk (Direct communication) • What if players can communicate before their actions? • If the players in prisoner’s dilemma can communicate, then? • If Harry and Sally can communicate in a battle of sexes game, then? • If players of asymmetric information can communicate, then?

  40. Cheap Talk are games with costlesscommunication which have no direct impact on the payoffs of the game. • Adding one more stage onto any game. • Babbling equilibrium : equilibria are independent of any communication.

  41. Prisoner’s dilemma • Battle of sexes (coordination games)

  42. When will the cheap talk be informative? • Prisoner’s dilemma 1. the receiver prefers different actions facing different types senders • Labor market/product quality 2. Senders of different types prefer different actions of the receivers

  43. Both Tough and Weak defenders (Senders) prefer the actions of NI by the Attacker (Receiver) and will both announce they’re Tough type.

  44. Communication with enemy/zero-sum game 3. The receiver’s preferences over actions cannot be completely opposed to the sender’s. (Players have common interests.)

More Related