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Signaling in matching markets

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  • 1. Signaling in Matching Markets Signaling in Matching Markets Peter Coles Alexey Kushnir Muriel Niederle Harvard Business School, The Pennsylvania State University, and Stanford University and NBER October 2009
  • 2. Signaling in Matching Markets Motivation Papers "Signaling in Matching Markets" by Peter Coles, Alexey Kushnir, and Muriel Niederle "Can Costless Signaling Be Harmful for Matching Markets?" Alexey Kushnir
  • 3. Signaling in Matching Markets Motivation Signaling in practice The entry-level market for clinical psychologists, Roth and Xing (1994) one-day market transactions by telephone College admissions, Avery and Levin (2009) early action and early decision colleges want to admit students who are enthusiastic about attending signal enthusiasm Electronic dating markets electronic roses Job market for new Ph.D. economists each candidate can send signals up to two departments signals are private
  • 4. Signaling in Matching Markets Motivation How signals can be helpful? Roth (2008) about the job market for new Ph.D. economists 1 Transmit information about Ph.D. candidate preferences transmit candidate speci…c preferences to departments: a candidate wants to obtain position in Europe/ on West or East U.S. coast/ in a speci…c city 2 Alleviate the coordination problem
  • 5. Signaling in Matching Markets Motivation When signals can be helpful? Congestion In congested markets there is limited period of time candidates may begin accepting o¤ers from other departments o¤ers are costly "...because of the arduous nature of the selection process, the hiring of one young professor can cost a school from $10,000 to $15,000." (Mark Whitehouse, The Wall Street Journal, January 8, 2007)
  • 6. Signaling in Matching Markets Motivation Main question Understand the value of a signaling mechanism for markets with various structures
  • 7. Signaling in Matching Markets Outline Outline 1 Literature review 2 A simple example 3 Model 4 Equilibrium analysis 5 Signals and agent welfare 6 Market structure and the value of a signaling mechanism 1 when is signaling most valuable? 2 optimal number of signals 3 many periods
  • 8. Signaling in Matching Markets Literature review Literature review Costless signaling Crawford and Sobel (1982) Sobel "Signaling games" (2009) Costless signaling in centralized matching markets Lee and Schwarz (2007) Abdulkadiroglu, Che, and Yasuda (2008) Costless signaling in decentralized matching markets Roth and Xing (1997) Avery and Levin (2009)
  • 9. Signaling in Matching Markets A simple example A simple example 2 …rms and 2 workers Preferences of …rms are i.i.d. and Pr (w1 w2 ) = Pr( w2 w1 ) = 1 fj fj 2 Preference of workers are i.i.d. and Pr (f1 1 w i f2 ) = Pr( f2 w i f1 ) = 2 Cardinal utility of agent a top choice ) 1 second choice ) x, 1 > x > 0 unmatched ) 0
  • 10. Signaling in Matching Markets A simple example Markets with signals Timing 1 Preferences are realized. Each worker sends up to one signal to one …rm. Workers send signals simultaneously. 2 Each …rm makes up to one o¤er to one worker. Firms make o¤ers simultaneously. 3 Each worker chooses an o¤er to accept among available o¤ers. Equilibrium concept: sequential equilibrium (with re…nement D1 of Cho and Kreps)
  • 11. Signaling in Matching Markets A simple example Markets with signals Workers strategies Proposition There are two types of symmetric sequential equilibria that satis…es criterion D1 Babbling equilibria Equilibria where workers send their signals to their top …rms
  • 12. Signaling in Matching Markets A simple example Markets with signals Firms strategies Firm’ tradeo¤: o¤er to better worker or o¤er to worker that s more likely to accept
  • 13. Signaling in Matching Markets A simple example Markets with signals Reduced game. Firm 1 receives a signal from its second choice Equilibria in pure strategies (respond, respond) is always an equilibrium respond=o¤er to 2nd choice if …rm 2 is responding, …rm 1 ignore=o¤er to 1st choice must respond! (ignore, ignore) is also an equilibrium if x < 0.5 …rm 1n…rm 2 respond ignore respond x x 1 ignore 0 2 1
  • 14. Signaling in Matching Markets A simple example Markets with signals Welfare (respond, respond) uf = 5 + 1 x, uw = 3 , µ = 7 (expected number of matches) 8 4 4 4 (ignore, ignore) uf = 3 , uw = 4 1 2 1 + 4 x, µ = 3 2
  • 15. Signaling in Matching Markets A simple example Observations from our simple example Firm strategies are strategic complements if …rm 1 responds more to signals, then …rm 2 is weakly better o¤ from responding more to signals Equilibrium ranking (ignore, ignore ) f (respond, respond ) (respond, respond ) w (ignore, ignore ) # of matches in (respond, respond ) > # of matches in (ignore, ignore )
  • 16. Signaling in Matching Markets A simple example Observations from our simple example Game with signals versus game without signals µsig µno_sig (uw )sig (uw )no_sig (uf )sig 7 (uf )no_sig
  • 17. Signaling in Matching Markets Model Model Model
  • 18. Signaling in Matching Markets Model Model F …rms, W workers Ordinal preferences θ f 2 ΘF …rm f ’ preference list (strict) s θ w 2 ΘW worker w ’ preference list (strict) s θ f and θ w are i.i.d. Cardinal utility of agent a ua ( , θ a ) > 0, consistent with θ a , ua (?, θ a ) = 0 for any permutation σ, ua (σ(θ f ), σ(w )) = ua (θ f , w )
  • 19. Signaling in Matching Markets Model Block-correlated preferences Block-correlated preferences
  • 20. Signaling in Matching Markets Model Block-correlated preferences Block-correlated preferences B blocks of …rms. Firm preferences are uniformly distributed: θ f U (Θf ), i.i.d. Workers’preferences are block uniform: 1 For any b < b 0 , where b, b 0 2 f1, ..., B g, each worker prefers every …rm in Fb to any …rm in Fb 0 . 2 Each worker’ preferences within block Fb are uniform and s uncorrelated.
  • 21. Signaling in Matching Markets Model Block-correlated preferences Agent strategies in the market with signals No-signal (no o¤er) option N Worker’ strategy is s sw : Θw ! ∆(F [ N ) last stage: worker w accepts the best o¤er Firm’ strategy s sf : Θ F 2W ! ∆(W [ N ) Equilibrium concept: sequential equilibrium (with re…nement D1 of Cho and Kreps)
  • 22. Signaling in Matching Markets Model Block-correlated preferences Assumptions De…nition Worker w ’ strategy sw is anonymous (neutral): s 8σ 2 Σ, θ w 2 Θw , σ(s (θ w )) = s (σ(θ w )). De…nition Firm f ’ strategy sf is anonymous (neutral): s 8σ 2 Σ, θ f 2 Θf , h W ) σ(s (θ f , h)) = s (σ(θ f ), σ(h)). where σ 2 Σ is some permutation of preference pro…les.
  • 23. Signaling in Matching Markets Equilibrium analysis Equilibrium analysis Equilibrium analysis
  • 24. Signaling in Matching Markets Equilibrium analysis Block-symmetric sequential equilibria De…nition Block-symmetric sequential equilibrium: Firms that are within each block use the same anonymous strategy and have the same beliefs. All workers use the same anonymous strategy.
  • 25. Signaling in Matching Markets Equilibrium analysis Block-symmetric sequential equilibria Characterization Proposition Let us consider some block-symmetric sequential equilibrium that satis…es criterion D1 of Cho and Kreps (1987). Then either 1 The equilibrium is a babbling equilibrium or 2 Workers use top-…rm strategies and …rms have top-…rm beliefs
  • 26. Signaling in Matching Markets Equilibrium analysis Block-symmetric sequential equilibria Babbling equilibria
  • 27. Signaling in Matching Markets Equilibrium analysis Block-symmetric sequential equilibria Top-…rms equilibria
  • 28. Signaling in Matching Markets Equilibrium analysis Block-symmetric sequential equilibria q b (p b ) are the ex-ante probability of receiving an o¤er from f 2 Fb , conditional on (not) sending a signal to …rm f . αb - the probability a worker sends her signal to block Fb . Proposition Let us consider some block-symmetric sequential equilibrium that satis…es criterion D1 of Cho and Kreps (1987). Then either 1 for any b 2 f1, ..., B g, q b = p b or 2 there exists b0 2 f1, ..., B g such that q b0 > p b0 and for any b : αb > 0, if a worker sends her signal to block Fb , she sends her signal to her most preferred …rm within Fb and qb > pb . for any b 0 : αb 0 = 0, the o¤-equilibrium beliefs of each …rm f 2 Fb 0 are such that µf (Γjw hf ) = 1, where Γ = fθ w 2 Θw : f = max (f 0 2 Fb 0 )g. θw
  • 29. Signaling in Matching Markets Equilibrium analysis Worker strategies We …x the worker strategies and …rms beliefs Workers play a symmetric, top-…rm strategy (α1 , ..., αB ). αb is the probability of sending a signal to top …rm within block Fb If …rm f receives worker w ’ signal its on- and o¤- equilibrium s beliefs are that it is top worker w ’ …rm within block F b . s We now examine stage 2 behavior
  • 30. Signaling in Matching Markets Equilibrium analysis Firm strategies Each …rm chooses between TSW and TRW
  • 31. Signaling in Matching Markets Equilibrium analysis Firm strategies: cuto¤ strategies
  • 32. Signaling in Matching Markets Equilibrium analysis Firm strategies: cuto¤ strategies Cuto¤ strategy is a vector (j1 , . . . , jW ) 2 [0, W ]W We have a natural partial order of cuto¤ strategies: j = (j1 , . . . , jW ) j 0 = j1 , . . . , jW , 8w = 1, ..., W , 0 0 jw 0 jw
  • 33. Signaling in Matching Markets Equilibrium analysis Firm strategies: cuto¤ strategies De…nition Strategy sf is a cuto¤ strategy for …rm f if 9j1 , . . . , jW 2 [1, W ] : for any θ f 2 Θf and h W , TSWf (θ f ) if rankθ f (TSWf (θ f )) jjh j s (θ f , h) = TRWf (θ f ) otherwise.
  • 34. Signaling in Matching Markets Equilibrium analysis Firm strategies: cuto¤ strategies Proposition (Optimality of Cuto¤ Strategies) For any strategy sf for …rm f , there exists a cuto¤ strategy which provides f weakly higher expected payo¤ than sf for any anonymous strategies s f of opponent …rms f .
  • 35. Signaling in Matching Markets Equilibrium analysis Existence Theorem There exists a block-symmetric sequential equilibrium where 1 workers play symmetric, top-…rm strategies and 2 …rms play block-symmetric, anonymous, cuto¤ strategies.
  • 36. Signaling in Matching Markets Signals and agent welfare Signals and agent welfare Signals and agent welfare
  • 37. Signaling in Matching Markets Signals and agent welfare Signals and agent welfare Proposition (Strategic complements) Suppose …rms f use cuto¤ strategies. If …rm f 0 2 f increases its cuto¤s (responds more to signals), …rm f will also optimally weakly increase its cuto¤s.
  • 38. Signaling in Matching Markets Signals and agent welfare Welfare comparison E (µ) No signaling Signaling E (W workers ) No signaling Signaling E (W …rms ) No signaling? Signaling
  • 39. Signaling in Matching Markets Signals and agent welfare One block of …rms: additional welfare results One block of …rms Additional welfare results One block of …rms: additional welfare results
  • 40. Signaling in Matching Markets Signals and agent welfare One block of …rms: additional welfare results One block of …rms One block of …rms Firm preferences, θ f U (Θf ), i.i.d. Worker preferences, θ w U ( Θw ), i.i.d.
  • 41. Signaling in Matching Markets Signals and agent welfare One block of …rms: additional welfare results Equilibrium existence Theorem There exists a symmetric sequential equilibrium in pure strategies where each worker send a signal to her top …rm and …rms employ symmetric strategies.
  • 42. Signaling in Matching Markets Signals and agent welfare One block of …rms: additional welfare results Welfare ranking of symmetric equilibria Proposition (Welfare ranking of symmetric equilibria) In a symmetric equilibrium with greater cuto¤s: the expected number of matches is higher workers have higher expected payo¤s …rms have lower expected payo¤s.
  • 43. Signaling in Matching Markets Extensions Extensions Many-to-many markets with many signals Workers can send several signals Firms have several positions to …ll additive valuations for any h W, uf (h, θ f ) = ∑w 2h uf (w , θ f ) Workers occupy several positions (ex. positions=interviews)
  • 44. Signaling in Matching Markets Market structure and the value of a signaling mechanism Market structure and the value of a signaling mechanism Market structure and the value of a signaling mechanism
  • 45. Signaling in Matching Markets Market structure and the value of a signaling mechanism Market structure and the value of a signaling mechanism Questions: Large vs small markets: when is signaling most valuable? Many periods of interactions What is the optimal number of signals?
  • 46. Signaling in Matching Markets Market structure and the value of a signaling mechanism Pure coordination model One block of …rms, B = 1 Agents care only about obtaining a match for any w 2 W , f 2 F , uw (f , θ w ) = uw > 0 for any w 2 W , f 2 F , uf (w , θ f ) = uf > 0
  • 47. Signaling in Matching Markets Market structure and the value of a signaling mechanism Balanced markets The value of a signaling mechanism D (F , W ) - the expected increase in the number of matches from the introduction of the signaling mechanism W=10 W=100 1.5 15 D D 1.0 10 0.5 5 0.0 0 0 10 20 30 40 50 0 100 200 300 400 500 F F F=10 F=100 1.5 15 D 1.0 D 10 0.5 5 0.0 0 0 10 20 30 40 50 0 100 200 300 400 500 W W
  • 48. Signaling in Matching Markets Market structure and the value of a signaling mechanism Balanced markets The value of a signaling mechanism for large markets Proposition (Large markets) D (F , W ) is "almost" a homogeneous of degree one D (F , W ) = F α( W ) + OF (1) F F D (F , W ) = W β( W ) + OW (1) where OF (1) and OW (1) are functions that are smaller than a constant for large F and W correspondingly. Proposition (Balanced markets) For …xed W , D (F , W ) attains its maximum value at F ' 1.0121W + OW (1). For …xed F , D (F , W ) attains its maximum value at W ' 1.8842F + OF (1).
  • 49. Signaling in Matching Markets Market structure and the value of a signaling mechanism Matching markets with many periods Matching markets with many periods There are L periods of interactions. Agents observe agents that match and leave the market. Period 0. Workers send their signals. Each worker sends one signal to some …rm. Periods 1, ..., L. Each period consists of two stages: Each …rm makes one o¤er to some worker. Each worker can accept one o¤er from the set of o¤ers she receives.
  • 50. Signaling in Matching Markets Market structure and the value of a signaling mechanism Matching markets with many periods Matching markets with many periods The symmetric sequential equilibrium in the o¤er game with signals each worker sends her signal to her top …rms; each worker accepts the best available o¤er immediately each …rm always responds to signals
  • 51. Signaling in Matching Markets Market structure and the value of a signaling mechanism Matching markets with many periods Matching markets with many periods Proposition In a market with F …rms and W workers, the value of a signaling mechanism D (F , W ) decreases with the number of periods of interactions.
  • 52. Signaling in Matching Markets Market structure and the value of a signaling mechanism Matching markets with many periods Matching markets with many periods Simulation results Simulation results: For the markets with F = W : D is decreasing over L. D decreases by 55% for L=2 D decreases by 95% for L=4
  • 53. Signaling in Matching Markets Market structure and the value of a signaling mechanism Matching markets with I interviews and K signals Matching markets with I interviews and K signals Simulation results Markets with I = 1 (solid), 2 (dashed), and 3(dot-dashed) interviews, W = 100 workers and F = I 100 …rms.
  • 54. Signaling in Matching Markets Summary Summary 1 Model of decentralized matching markets where signals transmit information alleviate coordination problem 2 We show that the introduction of a signaling mechanism 1 increases the expected number of matches 2 increases the expected welfare of workers 3 We analyze the value of a signaling mechanism for various market structures 1 Several signals 2 Several periods of interaction 3 Several …rms’positions 4 Several interviews
  • 55. Signaling in Matching Markets Summary Thank you
  • 56. Signaling in Matching Markets Motivating example for "Can private costless signaling be harmful for matching markets?" Example Motivating example for "Can costless signaling be harmful for matching markets?"
  • 57. Signaling in Matching Markets Motivating example for "Can private costless signaling be harmful for matching markets?" Model 3 …rms and 3 workers Preferences Firms’preferences are the same and publicly known θ f j = (w 1 , w 2 , w 3 ) Workers’preferences θ w i = (1 ε) θ 0 ε θ ai , i .i .d . θ 0 = (f1 , f2 , f3 ) the same and publicly known ("typical") θ ai U (Θw ) ("atypical")
  • 58. Signaling in Matching Markets Motivating example for "Can private costless signaling be harmful for matching markets?" Markets with signals 1 Preferences are realized. Each worker can send one signal to one …rm. 2 Each …rm can make one o¤er to one worker. 3 Each worker chooses an o¤er to accept among available o¤ers.
  • 59. Signaling in Matching Markets Motivating example for "Can private costless signaling be harmful for matching markets?" Matching market without signals Matching market without signals
  • 60. Signaling in Matching Markets Motivating example for "Can private costless signaling be harmful for matching markets?" Matching market with signals Matching market with signals
  • 61. Signaling in Matching Markets Motivating example for "Can private costless signaling be harmful for matching markets?" Observations from example In the o¤er game with signals Some workers received better matches Some workers (and …rms) are unmatched Firms, conditional on receiving a signal, obtain weakly better matches
  • 62. Signaling in Matching Markets Motivating example for "Can private costless signaling be harmful for matching markets?" Observations from example Game with signals versus game without signals µsig µno_sig (uw )sig 7 (uw )no_sig (uf )sig 7 (uf )no_sig

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