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Lesson overview

Lesson overview. Chapter 9 Uncertainty and Information Lesson II.3 Cheap Talk and Adverse Selection Theory Lesson II.4 Signaling and Screening Theory Each Example Game Introduces some Game Theory Example 1: Signaling and Screening Example 2: Pooling Example 3: Separating Equilibrium

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Lesson overview

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  1. Lesson overview • Chapter 9 Uncertainty and Information • Lesson II.3 Cheap Talk and Adverse Selection Theory • Lesson II.4 Signaling and Screening Theory • Each Example Game Introduces some Game Theory • Example 1: Signaling and Screening • Example 2: Pooling • Example 3: Separating Equilibrium • Example 4: Pooling Equilibrium • Example 5: Semiseparating Equilibrium • Lesson II.5 Signaling and Screening Applications BA 592 Lesson II.4 Signaling and Screening Theory

  2. Example 1: Signaling and Screening In games where cheap talk (costless communication) cannot be screened for information by the less informed player and cheap talk cannot signal information by the more informed player, costly communication can be used for screening and signaling. For example, employers know less about the skills and work attitude of potential employees than do the employees. Because unskilled and lazy potential employees will claim to have skills and will act hard working during interviews to get higher pay, employers cannot use costless communication to screen applicants. What about costly communication? Does that screen or signal information? BA 592 Lesson II.4 Signaling and Screening Theory

  3. Example 1: Signaling and Screening Suppose employers have two kinds of jobs. Skilled and hard working Type A (impatient, time-conscious, status-conscious, ambitious) employees can do Type A jobs, which contribute $160,000 per year to profits. And Type B (patient, easy-going, apathetic) employees can do Type B jobs, which contribute $60,000 per year to profits. Suppose Type A workers have existing jobs paying $125,000 per year, and Type B have jobs paying $30,000 per year. Suppose Type A workers regard the cost of completing a hard college class as $3,000 a year of salary, and Type B workers as $15,000 a year of salary. Suppose there are more jobs than workers. Determine whether the employer can use the differential cost of completing hard college courses to screen Type A workers from Type B workers. Or, whether workers can signal. BA 592 Lesson II.4 Signaling and Screening Theory

  4. Example 1: Signaling and Screening Consider finding an appropriate integer number N so the Employer should offer $160K per year for workers that have completed at least N courses, and $60K per year to the other workers. There are two constraints on N, incentive compatibility and participation. BA 592 Lesson II.4 Signaling and Screening Theory

  5. Example 1: Signaling and Screening Incentive compatibility constrains the number N of courses to be high enough so Type B workers do not bother to meet it, and low enough so Type A workers will meet it. Incentive compatibility for Type B requires $60,000 > $160,000 - $15,000 x N, or N > 6.67, meaning N > 7. Incentive compatibility for Type A requires $160,000 - $3,000 x N > $60,000, or N < 33.33, meaning N < 33. BA 592 Lesson II.4 Signaling and Screening Theory

  6. Example 1: Signaling and Screening Participation constrains the number N of courses so Type B and Type A workers both accept the job offers of $60,000 to Type B workers and $160,000 to Type A workers. Participation for Type B requires $60,000 > $30,000, which is true regardless of N. Participation for Type A requires $160,000 - $3,000 x N > $125,000, or N < 11.67, meaning N < 11. BA 592 Lesson II.4 Signaling and Screening Theory

  7. Example 1: Signaling and Screening Putting it all together, the Employer should pick any number between 7 and 11 and offer $160K per year for workers that have completed at least N courses, and $60K per year to the other workers. Those job offers separate Type A from Type B workers, but it comes at the communication cost of $3,000 x N per year paid by the Type A workers. Even if the smallest number of courses were picked (N = 7), the cost of the information asymmetry is $3,000 x 7 = $21,000 per year. BA 592 Lesson II.4 Signaling and Screening Theory

  8. Example 2: Pooling Dropping the screening by course selection avoids the $21,000 per year cost of the information asymmetry but means pooling all workers together, paying them the same. In particular, you overpay Type B workers. Determine whether the employer should use college courses to screen Type A workers from Type B workers, or whether the employer should pool workers. BA 592 Lesson II.4 Signaling and Screening Theory

  9. Example 2: Pooling Dropping the screening by course selection and pooling workers requires, in a competitive market, the common salary under pooling equals the population average of what the types are worth to the Employer, and that average depends on the proportions of the types in the population. On the one hand, if 60% of the population were Type A, and 40% Type B. Then the common salary is .6 x $160,000 + .4 x $60,000 = $120,000. Type B workers thus prefer the pooling equilibrium. Type A workers find the pooling salary $120,000 inferior to the $160,000-$21,000 = $139,000 net from screening. BA 592 Lesson II.4 Signaling and Screening Theory

  10. Example 2: Pooling On the other hand, if 80% of the population were Type A, and 20% Type B. Then the common salary is .8 x $160,000 + .2 x $60,000 = $140,000. Type B workers still prefer the pooling equilibrium to the screening equilibrium. And now Type A workers prefer the pooling equilibrium. However, that pooling equilibrium creates a Prisoner’s dilemma for Type A workers. Working with the 80%-20% of types, all Type A workers are best off by accepting the pooling equilibrium, but individually each worker could signal by offering to complete 1 course and work for $140,000 + $10,000 per year. BA 592 Lesson II.4 Signaling and Screening Theory

  11. Example 2: Pooling Employers would agree to pay $140,000 + $10,000 per year for a worker that completes 1 course because only Type A workers would be willing to take 1 course for an extra $10,000 per year. As more Type A workers take 1 course to signal their type, the average value of the pooled workers decreases, which increases the salary gap between Type A workers and pooled workers, which requires more courses to signal. This process continues until all Type A workers signal by taking 7 courses, and we are back at the equilibrium with separation of types based on screening (or signaling). BA 592 Lesson II.4 Signaling and Screening Theory

  12. Example 3: Separating Equilibrium Signaling and screening in the asymmetric information games above had many informed players and many uninformed players. Asymmetric information games with just two players allow the previous separating equilibrium and pooling equilibrium, and the new possibility of semiseparating equilibrium. BA 592 Lesson II.4 Signaling and Screening Theory

  13. Example 3: Separating Equilibrium Ford currently enjoys a monopoly in the market for the most economical-but-fun car, the Ford Mustang Convertible. Pontiac has a competing redesign of the Pontiac Solstice (to make it more economical) and is deciding whether to enter the market. But Pontiac does not know how tough a competitor Ford will prove to be. Specifically, Ford’s production cost is unknown to Pontiac, and may be high or low. If it is high, Pontiac can enter and compete profitably; if it is low, Pontiac’s entry and development costs cannot be recouped by subsequent operating profits, and it will make a net loss if it enters. BA 592 Lesson II.4 Signaling and Screening Theory

  14. Example 3: Separating Equilibrium The two firms interact in a three-stage sequential move game. In each stage, the per-unit costs and prices are in thousands of dollars, and the number of cars sold is in hundreds of thousands, so total costs and profits are in hundreds of millions. BA 592 Lesson II.4 Signaling and Screening Theory

  15. Example 3: Separating Equilibrium In the first stage, Nature chooses with probability probLow Ford to have Low cost cLow (thousand dollars) and with probability (1-probLow) Ford to have High cost cHigh. Ford observes Nature’s choice but Pontiac only knows only the probabilities and the possible cost values cLow and cHigh. In the second stage, Ford sets a price PLow or PHigh(thousand dollars) knowing that it is the only manufacturer in the market in Period 1, and that the selling price of cars depends on the total output of cars, P = 25-Q. Specifically, if Ford is Low cost cLow, it sets its monopoly price PLow. But if Ford is High cost cHigh, it chooses to either set its monopoly price PHigh or falsely signal Low cost by choosing PLow. BA 592 Lesson II.4 Signaling and Screening Theory

  16. Example 3: Separating Equilibrium In the third stage, Pontiac makes its entry decision in Period 2. To enter the market, Pontiac must pay 40 to build its plant, advertise, and so on. If Pontiac enters the market, its costs of producing and delivering each of its cars to the market is 10, and it becomes a Cournotduopolist with Ford, with the selling price of cars again depends on the total output of cars, P = 25-Q. BA 592 Lesson II.4 Signaling and Screening Theory

  17. Example 3: Separating Equilibrium Ford prefers Pontiac not enter the market. It might try to use its price in the second stage to signal a low cost. A low-cost firm would charge a lower Monopoly price than a high-cost firm. Pontiac is aware that Ford might falsely signal a low cost with a low price. Can Ford profitably signal low cost and keep Pontiac from entering? The answer depends on the probability probLowand cost cLowand cHighparameters. We consider three possible cases. BA 592 Lesson II.4 Signaling and Screening Theory

  18. Example 3: Separating Equilibrium • In the first case: • the probability that Ford is Low cost is probLow= 0.4 • the probability of High cost is (1-probLow) = 0.6 • the Low cost level is cLow= 5 • the High cost level is cHigh= 15 • Use those parameters to define the game tree for the Market Entry Game #1, and find all rollback solutions. BA 592 Lesson II.4 Signaling and Screening Theory

  19. Example 3: Separating Equilibrium First stage: Nature determines low unit cost of 5 for Ford with probability 0.4, and cost 15 with probability 0.6 Second stage: If Ford costs are Low, Ford sets the monopoly price, PLow=15 and earns 100. If costs are High, either set price PLow=15and earn 0, or set the monopoly price PHigh=20 and earn 25. Third stage: Pontiac Enters In or Stays Out. If Stays Out, Ford sets monopoly price according to its true costs, earning 100 if Ford’s costs are low or 25 if high. If Enters In, Ford and Pontiac are CournotDuopolists, earning 69.4 and 11.1 if Ford’s costs are low, or 2.7 and 44.4 if Ford’s costs are high. BA 592 Lesson II.4 Signaling and Screening Theory

  20. Example 3: Separating Equilibrium Partial rollback solution: If Ford prices High, then Pontiac Enters In. Remaining decisions: Ford prices Low or High if costs are High. Pontiac Enters In or Stays out if Ford chooses to price Low. In stage two, Ford’s two branches for Price Low are in the same information set for Pontiac, meaning Pontiac does not know which branch it is on when Ford prices Low. Pontiac is thus limited to choosing the same strategy (Enter In or Stay Out or a mixture) for each branch in the information set. BA 592 Lesson II.4 Signaling and Screening Theory

  21. Example 3: Separating Equilibrium Expected value of payoffs if Ford prices Low if costs are High and Pontiac Enters In if Ford chooses to price Low: 0.4(169,-29) + 0.6(3,4) = (69.4,-9.2) BA 592 Lesson II.4 Signaling and Screening Theory

  22. Example 3: Separating Equilibrium Expected value of payoffs if Ford prices Low if costs are High and Pontiac Stays Out if Ford chooses to price Low: 0.4(200,0) + 0.6(25,0) = (95,0) BA 592 Lesson II.4 Signaling and Screening Theory

  23. Example 3: Separating Equilibrium Expected value of payoffs if Ford prices High if costs are High and Pontiac Enters In if Ford chooses to price Low: 0.4(169,-29) + 0.6(28,4) = (84.4,-9.2) BA 592 Lesson II.4 Signaling and Screening Theory

  24. Example 3: Separating Equilibrium Expected value of payoffs if Ford prices High if costs are High and Pontiac Stays Out if Ford chooses to price Low: 0.4(200,0) + 0.6(28,4) = (96.8,2.4) The completed Normal form has High Price the dominate strategy for Ford, making Pontiac respond with Stay Out. That is the case pictured in the tree on the left. Ford signals its true type in the Separating Equilibrium. BA 592 Lesson II.4 Signaling and Screening Theory

  25. Example 4: Pooling Equilibrium • In the second case, cLow= 10 rather than cLow= 5: • the probability that Ford is Low cost is probLow= 0.4 • the probability of High cost is (1-probLow) = 0.6 • the Low cost level is cLow= 10 • the High cost level is cHigh= 15 • Use those parameters to define the game tree for the Market Entry Game #2, and find all rollback solutions. BA 592 Lesson II.4 Signaling and Screening Theory

  26. Example 4: Pooling Equilibrium First stage: Nature determines cLow= 10 for Ford with probability 0.4, and cost 15 with probability 0.6 Second stage: If Ford costs are Low, Ford sets the monopoly price, PLow=17.5 and earns 56. If costs are High, either set price PLow=17.5and earn 2.5x7.5 = 18.8, or set the monopoly price PHigh=20 and earn 25. Third stage: Pontiac Enters In or Stays Out. If Stays Out, Ford sets monopoly price according to its true costs, earning 56 if Ford’s costs are low or 25 if high. If Enters In, Ford and Pontiac are CournotDuopolists, earning 25 and 25 if Ford’s costs are low, or 2.7 and 44.4 if Ford’s costs are high. BA 592 Lesson II.4 Signaling and Screening Theory

  27. Example 4: Pooling Equilibrium Partial rollback solution: If Ford prices High, then Pontiac Enters In. Remaining decisions: Ford prices Low or High if costs are High. Pontiac Enters In or Stays out if Ford chooses to price Low. BA 592 Lesson II.4 Signaling and Screening Theory

  28. Example 4: Pooling Equilibrium Expected value of payoffs if Ford prices Low if costs are High and Pontiac Enters In if Ford chooses to price Low: 0.4(81,-15) + 0.6(22,4) = (45.6,-3.6) BA 592 Lesson II.4 Signaling and Screening Theory

  29. Example 4: Pooling Equilibrium Expected value of payoffs if Ford prices High if costs are High and Pontiac Enters In if Ford chooses to price Low: 0.4(81,-15) + 0.6(28,4) = (49.2,-3.6) BA 592 Lesson II.4 Signaling and Screening Theory

  30. Example 4: Pooling Equilibrium Expected value of payoffs if Ford prices High if costs are High and Pontiac Stays Out if Ford chooses to price Low: 0.4(112,0) + 0.6(28,4) = (61.6,2.4) BA 592 Lesson II.4 Signaling and Screening Theory

  31. Example 4: Pooling Equilibrium Expected value of payoffs if Ford prices Low if costs are High and Pontiac Stays Out if Ford chooses to price Low: 0.4(112,0) + 0.6(44,0) = (71.2,0) The completed Normal form has Stay Out the dominate strategy for Pontiac, making Pontiac respond with Price Low. That is the case pictured in the tree on the left. Ford signals Low Cost in the Pooling Equilibrium. BA 592 Lesson II.4 Signaling and Screening Theory

  32. Example 5: Semiseparating Equilibrium • In the third case, probLow= 0.1 rather than probLow= 0.4: • the probability that Ford is Low cost is probLow= 0.1 • the probability of High cost is (1-probLow) = 0.9 • the Low cost level is cLow= 10 • the High cost level is cHigh= 15 • Use those parameters to define the game tree for the Market Entry Game #3, and find all rollback solutions. BA 592 Lesson II.4 Signaling and Screening Theory

  33. Example 5: Semiseparating Equilibrium First stage: Nature determines cLow= 10 for Ford with probability 0.4, and cost 15 with probability 0.6 Second stage: If Ford costs are Low, Ford sets the monopoly price, PLow=17.5 and earns 56. If costs are High, either set price PLow=17.5and earn 2.5x7.5 = 18.8, or set the monopoly price PHigh=20 and earn 25. Third stage: Pontiac Enters In or Stays Out. If Stays Out, Ford sets monopoly price according to its true costs, earning 56 if Ford’s costs are low or 25 if high. If Enters In, Ford and Pontiac are CournotDuopolists, earning 25 and 25 if Ford’s costs are low, or 2.7 and 44.4 if Ford’s costs are high. BA 592 Lesson II.4 Signaling and Screening Theory

  34. Example 5: Semiseparating Equilibrium Partial rollback solution: If Ford prices High, then Pontiac Enters In. Remaining decisions: Ford prices Low or High if costs are High. Pontiac Enters In or Stays out if Ford chooses to price Low. Complete the normal form just as in Example 4, but with new probabilities. BA 592 Lesson II.4 Signaling and Screening Theory

  35. Example 5: Semiseparating Equilibrium Expected value of payoffs if Ford prices Low if costs are High and Pontiac Enters In if Ford chooses to price Low: 0.1(81,-15) + 0.9(22,4) = (27.9,2.1) Expected value of payoffs if Ford prices High if costs are High and Pontiac Enters In if Ford chooses to price Low: 0.1(81,-15) + 0.9(28,4) = (33.3,2.1) Expected value of payoffs if Ford prices High if costs are High and Pontiac Stays Out if Ford chooses to price Low: 0.1(112,0) + 0.9(28,4) = (36.4,3.6) Expected value of payoffs if Ford prices Low if costs are High and Pontiac Stays Out if Ford chooses to price Low: 0.1(112,0) + 0.9(44,0) = (50.8,0) BA 592 Lesson II.4 Signaling and Screening Theory

  36. Example 5: Semiseparating Equilibrium The completed Normal form has no Nash equilibrium in pure strategies. In mixed strategies, with probability p Ford prices Low if costs are High and with probability q Pontiac Stays Out if Ford chooses to price Low: Ford’s p-mix must make Pontiac indifferent between its pure strategies: 2.1p+2.1(1-p) = 0p+3.6(1-p), or p = 1.5/3.6 = .42 And Pontiac’s q-mix must make Ford indifferent between its pure strategies: 27.9q+50.8(1-q) = 33.3q+36.4(1-q), or q = 14.4/19.8 = .73 BA 592 Lesson II.4 Signaling and Screening Theory

  37. Example 5: Semiseparating Equilibrium Putting it all together, with probability p = .42 Ford prices Low if costs are High, and with q = .73 Pontiac Stays Out if Ford chooses to price Low. Ford’s types are thus only partially separated. The low cost type always prices low, but the high price type mixes and will also price low with 42% probability. A high price thus conveys full information, but a low price conveys only partial information about type in this Semiseparating Equilibrium. BA 592 Lesson II.4 Signaling and Screening Theory

  38. Lesson overview • Chapter 9 Uncertainty and Information • Lesson II.4 Signaling and Screening • Each Example Game Introduces some Game Theory • Example 1: Signaling and Screening • Example 2: Pooling • Example 3: Separating Equilibrium • Example 4: Pooling Equilibrium • Example 5: Semiseparating Equilibrium • Lesson II.1 Mixed Strategy Applications BA 592 Lesson II.4 Signaling and Screening Theory

  39. BA 592 Game Theory End of Lesson II.1 BA 592 Lesson II.4 Signaling and Screening Theory

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