1 / 23

Principles of Game Theory

Principles of Game Theory. Lecture 14: Signaling. Administrative. Homework due Saturday Last quiz on Sunday. Last time. Games of incomplete information Uncertainty over types of players Asymmetric information and the strategic manipulation of information

yelena
Download Presentation

Principles of Game Theory

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. Principles of Game Theory Lecture 14: Signaling

  2. Administrative • Homework due Saturday • Last quiz on Sunday

  3. Last time • Games of incomplete information • Uncertainty over types of players • Asymmetric information and the strategic manipulation of information • Today: Strategies for the more informed • Signaling • Signal Jamming • Next time: Strategies for the less informed • Incentive schemes • Screening devices

  4. Signaling • Signals: actions that more informed players use to convey information to less informed players about the unobservable type of the more informed player. • Often one player has information that he wants the other to know – “I’m a hard working type” • But often there are other types – e.g., “lazy types” – that also want to send the same message. • How does the “hardworking” type convey her message in such a way that the other player believes it? • Examples? • A player who wants the trust of less informed player may signal past instances of trust, may provide verbal assurances of trustworthiness, the names of character references/former employees on a resume, discuss church attendance, charity work, etc.

  5. Cheap Talk • We’ve already discussed a type of signaling games • When those actions are costless (or have a very low relative cost), they’re called “cheap talk” games • Signals may or may not be credible: Why? Because individuals will use signals strategically when it suits them. • Less qualified applicants may “pad” their resumes, lie about their past work history/qualifications, embezzle from their church/charity. • Talk is cheap: “Yeah, right”; “whatever”; “I could care less” are common. • The more credible signals involve costly actions, e.g. a college diploma, an artistic portfolio, a published book, a successful business.

  6. Signal Jamming • It’s the flip side of the problem: • How do “lazy” types conceal that they’re lazy given “hardworking” types take certain actions? • Often this involves the bad type playing a mixed strategy in order to confuse the other player.

  7. Market Entry Example • Assume we have two firms competing in a small tech industry and are trying to decide if they want to compete in a new sensor market: • Vecchio is well known but somewhat set-in-its-ways. Must decide if it wants to enter the new market • A startup, Newstar. Unknown startup from the Mideast. It’s a small company that only focuses on the small market • Vecchio believes Newstar could either be tough or weak competition, and Newstar knows this

  8. Market Entry • If both firms enter into the new market there will be a fight: • In a fight, Vecchio can beat a weak Newstar, but a tough Newstar can beat Vecchio. The winner has the market all to itself. • If Vecchio has the market to itself, it makes a profit of 3, and if Newstar has the market to itself it makes 4. • The cost of a fight is –2 to both firms.

  9. Market Entry • Equilibrium without signaling: • Let the probability that Newstar is “Weak” be w, and the probability of “Tough” is 1-w • What’s the Bayesian Nash equilibrium?

  10. Market Entry Eq w/o signaling • In the absence of any signals from Newstar, Vecchio will calculate the expected payoff from fighting, which is (w)1+(1-w)(-2)=3w-2, • Vecchiothencomparesit with the payoff from retreatingwhichis 0. • If3w-2 > 0, Vecchio’sbestresponseis to fight, or in otherwords, Vecchiofightsif: 3w > 2, orw > 2/3.  VecchiofightsonlyifitspriorbeliefisthatNewstarisverylikely to be weak, (chanceis 2 out of 3).

  11. Market Entry with Signaling • Suppose Newstar can provide a signal of its type by presenting some evidence that it is strong. • Say, by displaying prototypes of its new advanced products before it has the ability to produce and distribute a large quantity of these products. • Effects of signaling: • If it is unable to produce/distribute enough to meet market demand – if Newstar is “weak” – Vecchio may be able to copy the new products and quickly flood the market. • But if Newstar is “tough” and is ready to produce/distribute enough to meet market demand, it will squeeze Vecchio out of the market. • Newstar’s signal choice is therefore to display the new products, or not display the new products.

  12. Signaling with different costs • Suppose it is costly for a weak Newstar to imitate a strong Newstar. • Why? • A possible reason: A weak Newstar must hire more people/work overtime to have the products to display. • The cost for a weak Newstar to display, c, is common knowledge (along with w). The cost for a strong Newstar to display is 0. • What’s the game tree look like?

  13. Game tree Newstar: 0 Vecchio: 3 Newstar: 2 Vecchio: 0 Don’t Challenge Retreat Vecchio Newstar: -2 Vecchio: 1 Challenge & Don’t Display Newstar Fight “Weak” w Challenge & Display Retreat Newstar: 2-c Vecchio: 0 Vecchio Fight Newstar: -2-c Vecchio: 1 Nature Vecchio Retreat Newstar: 4 Vecchio: 0 1-w Challenge & Display Fight “Tough” Newstar Newstar: 2 Vecchio: -2 Don’t Challenge Newstar: 0 Vecchio: 3

  14. Market SignalingEq 1 • High cost signaling: • Suppose w = 0.5 (or w <2/3) and c = 3 (or, c>2) • Perfect Separation: • If Newstar Challenges and Displays, Vecchioknows Newstar is strong because it knows c>2 and can infer that only a strong Newstar would ever Challenge and Display, and so Vecchio always retreats in this case. • If Newstar is weak, and c>2, Newstar’s dominant strategy is not to challenge, because any challenge results in a negative payoff, even if Vecchio retreats. Newstar can get a 0 payoff from not challenging, so it does.

  15. Market SignalingEq 2 • Low cost signaling: • Now Suppose w = 0.5 (or w <2/3) and c = 1 (or c<2) • Pooling: • Both types of Newstars find it profitable to Challenge and Display because Vecchio will retreat– a pooling equilibrium.

  16. Market SignalingEq 3 • Low cost signaling and Weak types more likely: • Now Suppose w = 0.75 (or w >2/3) and c = 1 (or c<2) • No complete pooling or complete separating eq:

  17. No full separation • Why no separating equilibrium? • Suppose weak Newstar plays don’t challenge and strong Newstar plays challenge and display (Note: Challenge and don’t display doesn’t make sense because Vecchio will fight): • Vecchioretreats when it sees challenge and display. Weak Newstar will deviate. • Suppose weak Newstar plays challenge and display and strong Newstar plays don’t challenge: • Vecchio fights when it sees challenge and display. Weak Newstar will deviate to don’t challenge.

  18. No complete pooling • Why no pooling equilibrium? • If Vecchio sees challenge and display it Fights. Why? Thus weak Newstar deviates to don’t challenge.

  19. Semi-Separating • Now we have to consider mixed strategies: • Suppose weak Newstar chooses challenge and display with some probability pand doesn’t challenge with probability 1-p. • i.e., P(challenge & display | Weak) = p • Vecchio responds to a display by fighting with probability q. • Strong Newstar always challenges and displays. • Note: Vecchio draws inferences conditionalon whether or not Newstar displays.

  20. How does a smart Vecchio react? • Vecchio must form beliefs according to Bayes Rule! • If Vecchiosees a display, with what probability does it believe Newstar is weak? • wp/(1-w+wp) • Is strong? • (1-w)/(1-w+wp)

  21. Mixed strategies • Now we must find a mixed strategy (p and q) to play players indifferent – with the updated beliefs • Vecchio’s expected payoff from fighting conditional on observing a display is now: 1(wp/(1-w+wp) + (-2)[(1-w)/(1-w+wp)] =[wp-2(1-w)]/(1-w+wp) • Vecchio’s (expected) payoff from retreating is remains 0. • So, in mixing, Newstar chooses a p so as to keep Vecchio perfectly indifferent between fighting (1) and retreating (2): [wp-2(1-w)]/(1-w+wp)=0 or[wp-2(1-w)]=0 p=2(1-w)/w.

  22. Mixed Strategies continued • Given Vecchio’s strategy of fighting when it sees a display with probability q, a weak Newstar’s expected payoff from challenging with a display is: q(-2-c)+(1-q)(2-c)=2-c-4q • A weak Newstar’s (expected) payoff from not challenging is always 0. • So, Vecchio chooses q to keep a weak Newstar perfectly indifferent between challenging with display and not challenging 2-c-4q=0 q=(2-c)/4. • Summary: Mixed strategy, semi-separating equilibrium is for weak Newstar, to display with probability p=2(1-w)/w, and for Vecchio to challenge with probability q=(2-c)/4.

  23. Market Entry with Signaling:Summary • We can summarize the equilibria:

More Related