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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

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Principles of Game Theory

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  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:

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