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Not So Cheap Talk: Costly and Discrete Communication

Not So Cheap Talk: Costly and Discrete Communication. John Smith Rutgers University-Camden w/ Johanna Hertel. Introduction. Reality is complex and nuanced Often use words to communicate information about reality Single word can rarely convey entire extent of the complex reality

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Not So Cheap Talk: Costly and Discrete Communication

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  1. Not So Cheap Talk: Costly and Discrete Communication John Smith Rutgers University-Camden w/ Johanna Hertel

  2. Introduction • Reality is complex and nuanced • Often use words to communicate information • about reality • Single word can rarely convey • entire extent of the complex reality • More “things” • than “words” • Can approach desired complexity • by constructing more elaborate and complex statements • comes at a cost to the sender

  3. Tyson Hartwig • Coauthor of experimental paper • Undergrad who • now at UBC grad school • Our paper accepted at Stony Brook • I wanted feedback • I could not attend • Tyson could • I know much more about • theory and • presenting

  4. Tyson Hartwig • We have identical preferences over his performance • Effectively communicate • Look competent • Help get him into grad school • Q: Why didn’t I communicate • EVERYTHING relevant to the presentation? • A: Communication is costly

  5. Cheap Talk • Crawford and Sobel (1982) • Informed sender • Unformed receiver • Sender to take action which affects both • Communication is costless • Different preferences over action • Equilibria where meaningful communication

  6. Related Literature: Cheap Talk • Morgan and Stocken (2003) • Uncertainty divergence of preferences • Fischer and Stocken (2001) • Sender imperfect information about state • Blume, Board and Kawamura (2007) • Communication errors • Same spirit as these

  7. Related Literature: NITS • Many equilibria in Crawford and Sobel • Chen, Kartik and Sobel (2008) • Impose additional restriction on CS equilibrium • No Incentive to Separate (NITS) • Only most informative equilibrium satisfies condition

  8. Related Literature: Costly Communication • Austen-Smith and Banks (2000) • Kartik (2007) • Add availability of costly messages to CS • Expand set of equilibria • Dewatripont and Tirole (2005) • Austen-Smith (1994) • Information understood or not

  9. Related Literature: Costly Communication • Cremer, Garicano and Prat (2007) • Finite number of partition elements • Sender and Receiver with aligned preferences • Optimal allocation into partitions • To coordinate

  10. Model • State of the world: s • uniformly distributed on [0,1] • exact value known by sender • unknown to receiver • Receiver to take action: a • on real numbers • Receiver's Payoffs: uR(a,s)=-(a-s)2

  11. Message Space • Sender can send messages • Message space: M ={m0,m1,m2,…} • Cost of messages • Message mi costs: c(i) • if i>j>0 • c(i)>c(j)>c(0)=0 • c(i)-c(i-1)>0 • Sender’s Payoffs: uS(a,s)=-(a-s-b)2-c(i) where b0

  12. Strategies • Sender’s Strategy: :[0,1]M • Receiver’s Strategy: : M R

  13. Model: Equilibrium Conditions • Equilibrium (*, *) require: • Receiver selects optimal action given message • Sender sends optimal message • Beliefs are derived from Bayes’ Rule whenever possible

  14. Necessary Conditions • h interval on which the hth message is sent • mh= *(s) for s[sh,sh+1) • Two relatively obvious necessary conditions: • 1+2+…+n=1 • where h0 for every h{1,…,n}

  15. Another Necessary Condition • h interval on which the hth message is sent • i interval size with message mi • Lemma 1: Necessary condition for equilibrium (h+1j)2–(hi)2 =4[c(i)-c(j)]+4b(h+1j+hi) • Messages associated with higher states • Are sent on larger intervals • Same as Crawford and Sobel (1982) • More costly signals are conserved • Are sent on smaller intervals

  16. Another Necessary Condition • Lemma 1: Necessary condition for equilibrium (h+1j)2–(hi)2 =4[c(i)-c(j)]+4b(h+1j+hi) • c=0 and b>0 • Costless communication, imperfect alignment • Reduces to (21) in Crawford and Sobel (1982) • c>0 and b=0 • Costly communication, perfect alignment • Order of intervals does not matter • Costly signals are conserved

  17. Equilibrium Characterization • What can we say without any restrictions of out-of-equilibrium beliefs? • Lemma 2: If c>0, then for any equilibrium • cheapest message (m0) will be transmitted for some states • and • will exist a completely uninformative equilibrium • cheapest message (m0) will be transmitted on all states

  18. Definition of Feasible • A (, ) is feasible if • satisfies necessary conditions • empty message is transmitted for some states

  19. Equilibrium • Proposition 1: All (, ) which are feasible • Can form an equilibrium • There are many, many equilibria • Like Crawford and Sobel

  20. c(i)=0.01(i) b=0 Eq. without “holes” Example Also, all permutations 0 1 1 m0 1 2 m0 m1 0.52 0.48 3 m0 m1 m2 0.271 0.392 0.337 4 m0 m1 m2 m3 0.336 0.302 0.227 0.108 No equilibrium with more than 4 messages

  21. c(i)=0.01(i) b=0 Eq. with “holes” Example Also, all permutations 0 1 m0 m1 0.52 0.48 m0 m2 0.46 0.54 m0 m3 2 0.56 0.44 . . . m0 m24 0.98 0.02 Many such equilibria involving 3 messages

  22. Restrict out-of-equilibrium beliefs • Which equilibria are admitted • if we restrict out-of-equilibrium beliefs? • No incentive to separate (NITS) • If observe out-of-equilibrium message • Receiver believes s=0

  23. s uS c(i)=0.02i b=0

  24. s uS c(i)=0.02i b=0 m0

  25. s uS c(i)=0.02i b=0 m0 m1

  26. s uS c(i)=0.02i b=0 m0 m1 m2

  27. s uS c(i)=0.02i b=0 m0 m1 m2

  28. s uS c(i)=0.02i b=0 m0 m1 m2

  29. s uS c(i)=0.02i b=0.01

  30. s uS c(i)=0.02i b=0.01 m0

  31. s uS c(i)=0.02i b=0.01 m0 m1

  32. s uS c(i)=0.02i b=0.01 And similarly for c=0 m0 m1 m2

  33. s uS c=0 b=0.01

  34. The Reasonableness of NITS • Is NITS reasonable? • Is NITS more reasonable when b=0?

  35. NITS Discussion • Chen, Kartik, and Sobel (2008) apply NITS • uniform-quadratic case Crawford Sobel (1982) • only equilibrium with maximum partitions • Our specification is not identical

  36. Perfect Alignment of Preferences (b=0) • Things work pretty well • if b=0, c>0 • Condition NITS, picks out • “best” possible equilibria

  37. Results: No Holes • Lemma 3: Consider equilibrium with mi • Under Condition NITS, • if b=0 then every mj where ji • is also used in equilibrium • Intuition: Will not use a more costly signal • when a less costly one is unused

  38. Definition: NITS-Feasible • An (, ) is NITS-feasible if • satisfies necessary conditions • Lemma 3 is satisfied (“No Holes”)

  39. Maximal, NITS-feasible • By “No Holes” result • can characterize NITS-feasible (k, k) • by the most costly message mk • We will say that (k, k) maximal, NITS-feasible if • does not exist NITS-feasible (k’, k’) for k’>k • Lemma 4: If b=0 and c>0 there always exists • maximal, NITS-feasible (, )

  40. Existence and Uniqueness • Proposition 2: If b=0 • then under Condition NITS, • an equilibrium always exists • and it is exclusively • the maximal, NITS-feasible class

  41. Discussion: Existence and Uniqueness • Condition NITS • b=0 and c>0, uniform-quadratic • Unique: only most informative equilibria • Analogous to NITS • Chen, Kartik and Sobel (2008)) applied to original cheap talk model • b>0 and c=0, uniform-quadratic • Unique: only most informative equilibrium • When b>0 and c>0 • Not guaranteed uniqueness

  42. Only maximal, NITS-feasible Example c(i)=0.01(i) b=0 NITS 0 1 1 Not admitted m0 2 m0 Not admitted m1 3 Not admitted m0 m1 m2 4 m0 m1 m2 m3 No admissible equilibrium 5 or more messages

  43. Suppose that b=0.245 c(i)=0.01*i Equilibrium with single message m0 on [0,1] Equilibrium with two messages m0 on [0,0.03) m1 on [0.03,1] Example: Nonuniqueness when b>0

  44. Nonuniqueness Intuition • Q: Why does uniqueness fail for • b>0 and c>0? • For b>0 and c=0 • Intervals increasing along state space • For b=0 and c>0 • Intervals exclusively determined by costs • But when b>0 and c>0 • These effects interact • Can cause nonuniqueness

  45. Suppose that b=0.2 c(i)=0.01*i Two equilibria m1 on [0,0.08) m0 on [0.08,1] And m0 on [0,0.12) m1 on [0.12,1] Now suppose that c(i)=0.02*i One of two equilibria m0 on [0,0.14) m1 on [0.14,1] Better communication here! outperforms Goltsman et al. (2009) JET upper bound for receiver’s payoffs Example: Communication Costs

  46. Intuition 0 1 m1  • Intervals on higher numbers • larger • Intervals involving costly signals • smaller 0.06 0.94  m1 0.14 0.86

  47. Conclusion • Communication game: • Communication costs increasing • in complexity of message sent • Shades of understanding • When b=0, Condition NITS • unique class of equilibria with most messages

  48. Alternate Specifications of Communication • Q: Why not specify communication costs • which are decreasing in the size of interval? • A1: Pooling equilibrium seems more natural • Not a unique word for every object • A2: Some noise is helpful to communication • Blume, Board and Kawamura (2007) • A3: Must assume that receiver is • not sophisticated

  49. Setup: Communication seems to work this way Reality complex and nuanced Communication discrete and costly Natural way to vary cost of communication Results: “Some” communication when preferences are aligned (b=0) shades of understanding Pooling equilibrium When b>0 and c>0 Can outperform Goltsman et al. (2009) bound “So Why Should I Care?”

  50. Experimental • Experiments in communication games • Cai and Wang (2006) • Kawagoe and Takizawa (2009) • Test with simplified model

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