{"history":[],"article":{"id":"6ac602d3-fdf3-4d4d-b517-de82d449e13a","title":"Cheap talk","slug":"cheap-talk","content":":::summary\n:::\n\n# \nIn [game theory](https://bliptext.com/articles/game-theory), **cheap talk** is communication between players that does not directly affect the payoffs of the game.\nProviding and receiving information is free.\nThis is in contrast to [signalling](https://bliptext.com/articles/signalling-economics), in which sending certain messages may be costly for the sender depending on the state of the world.\nThis basic setting set by [Vincent Crawford](https://bliptext.com/articles/vincent-crawford) and [Joel Sobel](https://bliptext.com/articles/joel-sobel) has given rise to a variety of variants.\nTo give a formal definition, cheap talk is communication that is:\nTherefore, an agent engaging in cheap talk could lie with impunity, but may choose in equilibrium not to do so.\n\n# Applications\n\n## Game theory\nCheap talk can, in general, be added to any game and has the potential to enhance the set of possible equilibrium outcomes.\nFor example, one can add a round of cheap talk in the beginning of the [Battle of the Sexes](https://bliptext.com/articles/battle-of-the-sexes-game-theory).\nEach player announces whether they intend to go to the football game, or the opera.\nBecause the Battle of the Sexes is a [coordination game](https://bliptext.com/articles/coordination-game), this initial round of communication may enable the players to select among multiple equilibria, thereby achieving higher payoffs than in the uncoordinated case.\nThe messages and strategies which yield this outcome are symmetric for each player.\nThey are: 1) announce opera or football with even probability 2) if a person announces opera (or football), then upon hearing this message the other person will say opera (or football) as well (Farrell and [Rabin](https://bliptext.com/articles/matthew-rabin), 1996).\nIf they both announce different options, then no coordination is achieved.\nIn the case of only one player messaging, this could also give that player a first-mover advantage.\nIt is not guaranteed, however, that cheap talk will have an effect on equilibrium payoffs.\nAnother game, the [Prisoner's Dilemma](https://bliptext.com/articles/prisoner-s-dilemma), is a game whose only equilibrium is in dominant strategies.\nAny pre-play cheap talk will be ignored and players will play their dominant strategies (Defect, Defect) regardless of the messages sent.\n\n## Biological applications\nIt has been commonly argued that cheap talk will have no effect on the underlying structure of the game.\nIn [biology](https://bliptext.com/articles/biology) authors have often argued that costly signalling best explains signalling between animals (see [Handicap principle](https://bliptext.com/articles/handicap-principle), [Signalling theory](https://bliptext.com/articles/signalling-theory)).\nThis general belief has been receiving some challenges (see work by Carl Bergstrom and [Brian Skyrms](https://bliptext.com/articles/brian-skyrms) 2002, 2004).\nIn particular, several models using [evolutionary game theory](https://bliptext.com/articles/evolutionary-game-theory) indicate that cheap talk can have effects on the evolutionary dynamics of particular games.\n\n# Crawford and Sobel's definition\n\n## Setting\nIn the basic form of the game, there are two players communicating, one sender S and one receiver R.\n\n### Type\nSender S gets knowledge of the state of the world or of his \"type\" t.\nReceiver R does not know t ; he has only ex-ante beliefs about it, and relies on a message from S to possibly improve the accuracy of his beliefs.\n\n### Message\nS decides to send message m.\nMessage m may disclose full information, but it may also give limited, blurred information: it will typically say \"The state of the world is between t1 and t2\".\nIt may give no information at all.\nThe form of the message does not matter, as long as there is mutual understanding, common interpretation.\nIt could be a general statement from a central bank's chairman, a political speech in any language, etc. Whatever the form, it is eventually taken to mean \"The state of the world is between t1 and t2\".\n\n### Action\nReceiver R receives message m.\nR updates his beliefs about the state of the world given new information that he might get, using [Bayes's rule](https://bliptext.com/articles/bayes-rule).\nR decides to take action a.\nThis action impacts both his own utility and the sender's utility.\n\n### Utility\nThe decision of S regarding the content of m is based on maximizing his utility, given what he expects R to do.\nUtility is a way to quantify satisfaction or wishes.\nIt can be financial profits, or non-financial satisfaction—for instance the extent to which the environment is protected.\n→ Quadratic utilities:\nThe respective utilities of S and R can be specified by the following:\nThe theory applies to more general forms of utility, but quadratic preferences makes exposition easier.\nThus S and R have different objectives if b ≠ 0.\nParameter b is interpreted as conflict of interest between the two players, or alternatively as bias.UR is maximized when a = t, meaning that the receiver wants to take action that matches the state of the world, which he does not know in general.\nUS is maximized when a = t + b, meaning that S wants a slightly higher action to be taken, if b > 0.\nSince S does not control action, S must obtain the desired action by choosing what information to reveal.\nEach player's utility depends on the state of the world and on both players' decisions that eventually lead to action a.\n\n### Nash equilibrium\nWe look for an equilibrium where each player decides optimally, assuming that the other player also decides optimally.\nPlayers are rational, although R has only limited information.\nExpectations get realized, and there is no incentive to deviate from this situation.\n\n## Theorem\nCrawford and Sobel characterize possible [Nash equilibria](https://bliptext.com/articles/nash-equilibria).\nWhen interests are aligned, then information is fully disclosed.\nWhen conflict of interest is very large, all information is kept hidden.\nThese are extreme cases.\nThe model allowing for more subtle case when interests are close, but different and in these cases optimal behavior leads to some but not all information being disclosed, leading to various kinds of carefully worded sentences that we may observe.\nMore generally:\n# [Figure 1: Cheap talk communication setting | upload.wikimedia.org/wikipedia/commons/6/6a/Cheap///talk.png]\n\n### Messages\nWhile messages could ex-ante assume an infinite number of possible values μ(t) for the infinite number of possible states of the world t, actually they may take only a finite number of values (m1, m2, . . ., mN).\nThus an equilibrium may be characterized by a partition (t0(N), t1(N). . . tN(N)) of the set of types [0, 1],\nwhere ''0 = t0(N) < t1(N) <.\n. . < tN(N) = 1''.\nThis partition is shown on the top right segment of Figure 1.\nThe ti(N)'s are the bounds of intervals where the messages are constant: for ti-1(N) < t < ti(N), μ(t) = mi.\n\n### Actions\nSince actions are functions of messages, actions are also constant over these intervals:\nfor ti-1(N) < t < ti(N), α(t) = α(mi) = ai.\nThe action function is now indirectly characterized by the fact that each value ai optimizes return for the R, knowing that t is between t1 and t2.\nMathematically (assuming that t is uniformly distributed over [0, 1]),\n→ Quadratic utilities:\nGiven that R knows that t is between ti-1 and ti, and in the special case quadratic utility where R wants action a to be as close to t as possible, we can show that quite intuitively the optimal action is the middle of the interval:\n\n### Indifference condition\nAt\n, The sender has to be indifferent between sending either message\nmi-1\nor\nmi\n. 1 ≤ i≤ N-1\nThis gives information about N and the ti.\n→ Practically:\nWe consider a partition of size N.\nOne can show that\nN must be small enough so that the numerator is positive.\nThis determines the maximum allowed value where is the ceiling of Z, i.e. the smallest positive integer greater or equal to Z. Example: We assume that b = 1/20.\nThen N* = 3.\nWe now describe all the equilibria for N=1, 2, or 3 (see Figure 2).\n**N = 1:** This is the babbling equilibrium.\nt0 = 0, t1 = 1; a1 = 1/2 = 0.5.\n**N = 2:** t0 = 0, t1 = 2/5 = 0.4, t2 = 1; a1 = 1/5 = 0.2, a2 = 7/10 = 0.7.\n**N = N* = 3:** t0 = 0, t1 = 2/15, t2 = 7/15, t3 = 1; a1 = 1/15, a2 = 3/10 = 0.3, a3 = 11/15.\nWith N = 1, we get the coarsest possible message, which does not give any information.\nSo everything is red on the top left panel.\nWith N = 3, the message is finer.\nHowever, it remains quite coarse compared to full revelation, which would be the 45° line, but which is not a Nash equilibrium.\nWith a higher N, and a finer message, the blue area is more important.\nThis implies higher utility.\nDisclosing more information benefits both parties.\n# [Figure 2: Message and utilities for conflict of interest b = 1/20, for N=1, 2, and 3 | upload.wikimedia.org/wikipedia/commons/9/91/Crawford///Sobel.png]","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T21:21:28.826Z","updated_at":"2025-01-25T21:21:28.826Z","search_vector":"'cheap':1 'talk':2","search_content":" \n \n\n# \nIn [game theory](https://bliptext.com/articles/game-theory), **cheap talk** is communication between players that does not directly affect the payoffs of the game.\nProviding and receiving information is free.\nThis is in contrast to [signalling](https://bliptext.com/articles/signalling-economics), in which sending certain messages may be costly for the sender depending on the state of the world.\nThis basic setting set by [Vincent Crawford](https://bliptext.com/articles/vincent-crawford) and [Joel Sobel](https://bliptext.com/articles/joel-sobel) has given rise to a variety of variants.\nTo give a formal definition, cheap talk is communication that is:\nTherefore, an agent engaging in cheap talk could lie with impunity, but may choose in equilibrium not to do so.\n\n# Applications\n\n## Game theory\nCheap talk can, in general, be added to any game and has the potential to enhance the set of possible equilibrium outcomes.\nFor example, one can add a round of cheap talk in the beginning of the [Battle of the Sexes](https://bliptext.com/articles/battle-of-the-sexes-game-theory).\nEach player announces whether they intend to go to the football game, or the opera.\nBecause the Battle of the Sexes is a [coordination game](https://bliptext.com/articles/coordination-game), this initial round of communication may enable the players to select among multiple equilibria, thereby achieving higher payoffs than in the uncoordinated case.\nThe messages and strategies which yield this outcome are symmetric for each player.\nThey are: 1) announce opera or football with even probability 2) if a person announces opera (or football), then upon hearing this message the other person will say opera (or football) as well (Farrell and [Rabin](https://bliptext.com/articles/matthew-rabin), 1996).\nIf they both announce different options, then no coordination is achieved.\nIn the case of only one player messaging, this could also give that player a first-mover advantage.\nIt is not guaranteed, however, that cheap talk will have an effect on equilibrium payoffs.\nAnother game, the [Prisoner's Dilemma](https://bliptext.com/articles/prisoner-s-dilemma), is a game whose only equilibrium is in dominant strategies.\nAny pre-play cheap talk will be ignored and players will play their dominant strategies (Defect, Defect) regardless of the messages sent.\n\n## Biological applications\nIt has been commonly argued that cheap talk will have no effect on the underlying structure of the game.\nIn [biology](https://bliptext.com/articles/biology) authors have often argued that costly signalling best explains signalling between animals (see [Handicap principle](https://bliptext.com/articles/handicap-principle), [Signalling theory](https://bliptext.com/articles/signalling-theory)).\nThis general belief has been receiving some challenges (see work by Carl Bergstrom and [Brian Skyrms](https://bliptext.com/articles/brian-skyrms) 2002, 2004).\nIn particular, several models using [evolutionary game theory](https://bliptext.com/articles/evolutionary-game-theory) indicate that cheap talk can have effects on the evolutionary dynamics of particular games.\n\n# Crawford and Sobel's definition\n\n## Setting\nIn the basic form of the game, there are two players communicating, one sender S and one receiver R.\n\n### Type\nSender S gets knowledge of the state of the world or of his \"type\" t.\nReceiver R does not know t ; he has only ex-ante beliefs about it, and relies on a message from S to possibly improve the accuracy of his beliefs.\n\n### Message\nS decides to send message m.\nMessage m may disclose full information, but it may also give limited, blurred information: it will typically say \"The state of the world is between t1 and t2\".\nIt may give no information at all.\nThe form of the message does not matter, as long as there is mutual understanding, common interpretation.\nIt could be a general statement from a central bank's chairman, a political speech in any language, etc. Whatever the form, it is eventually taken to mean \"The state of the world is between t1 and t2\".\n\n### Action\nReceiver R receives message m.\nR updates his beliefs about the state of the world given new information that he might get, using [Bayes's rule](https://bliptext.com/articles/bayes-rule).\nR decides to take action a.\nThis action impacts both his own utility and the sender's utility.\n\n### Utility\nThe decision of S regarding the content of m is based on maximizing his utility, given what he expects R to do.\nUtility is a way to quantify satisfaction or wishes.\nIt can be financial profits, or non-financial satisfaction—for instance the extent to which the environment is protected.\n→ Quadratic utilities:\nThe respective utilities of S and R can be specified by the following:\nThe theory applies to more general forms of utility, but quadratic preferences makes exposition easier.\nThus S and R have different objectives if b ≠ 0.\nParameter b is interpreted as conflict of interest between the two players, or alternatively as bias.UR is maximized when a = t, meaning that the receiver wants to take action that matches the state of the world, which he does not know in general.\nUS is maximized when a = t + b, meaning that S wants a slightly higher action to be taken, if b > 0.\nSince S does not control action, S must obtain the desired action by choosing what information to reveal.\nEach player's utility depends on the state of the world and on both players' decisions that eventually lead to action a.\n\n### Nash equilibrium\nWe look for an equilibrium where each player decides optimally, assuming that the other player also decides optimally.\nPlayers are rational, although R has only limited information.\nExpectations get realized, and there is no incentive to deviate from this situation.\n\n## Theorem\nCrawford and Sobel characterize possible [Nash equilibria](https://bliptext.com/articles/nash-equilibria).\nWhen interests are aligned, then information is fully disclosed.\nWhen conflict of interest is very large, all information is kept hidden.\nThese are extreme cases.\nThe model allowing for more subtle case when interests are close, but different and in these cases optimal behavior leads to some but not all information being disclosed, leading to various kinds of carefully worded sentences that we may observe.\nMore generally:\n# [Figure 1: Cheap talk communication setting | upload.wikimedia.org/wikipedia/commons/6/6a/Cheap///talk.png]\n\n### Messages\nWhile messages could ex-ante assume an infinite number of possible values μ(t) for the infinite number of possible states of the world t, actually they may take only a finite number of values (m1, m2, . . ., mN).\nThus an equilibrium may be characterized by a partition (t0(N), t1(N). . . tN(N)) of the set of types [0, 1],\nwhere ''0 = t0(N) < t1(N) <.\n. . < tN(N) = 1''.\nThis partition is shown on the top right segment of Figure 1.\nThe ti(N)'s are the bounds of intervals where the messages are constant: for ti-1(N) < t < ti(N), μ(t) = mi.\n\n### Actions\nSince actions are functions of messages, actions are also constant over these intervals:\nfor ti-1(N) < t < ti(N), α(t) = α(mi) = ai.\nThe action function is now indirectly characterized by the fact that each value ai optimizes return for the R, knowing that t is between t1 and t2.\nMathematically (assuming that t is uniformly distributed over [0, 1]),\n→ Quadratic utilities:\nGiven that R knows that t is between ti-1 and ti, and in the special case quadratic utility where R wants action a to be as close to t as possible, we can show that quite intuitively the optimal action is the middle of the interval:\n\n### Indifference condition\nAt\n, The sender has to be indifferent between sending either message\nmi-1\nor\nmi\n. 1 ≤ i≤ N-1\nThis gives information about N and the ti.\n→ Practically:\nWe consider a partition of size N.\nOne can show that\nN must be small enough so that the numerator is positive.\nThis determines the maximum allowed value where is the ceiling of Z, i.e. the smallest positive integer greater or equal to Z. Example: We assume that b = 1/20.\nThen N* = 3.\nWe now describe all the equilibria for N=1, 2, or 3 (see Figure 2).\n**N = 1:** This is the babbling equilibrium.\nt0 = 0, t1 = 1; a1 = 1/2 = 0.5.\n**N = 2:** t0 = 0, t1 = 2/5 = 0.4, t2 = 1; a1 = 1/5 = 0.2, a2 = 7/10 = 0.7.\n**N = N* = 3:** t0 = 0, t1 = 2/15, t2 = 7/15, t3 = 1; a1 = 1/15, a2 = 3/10 = 0.3, a3 = 11/15.\nWith N = 1, we get the coarsest possible message, which does not give any information.\nSo everything is red on the top left panel.\nWith N = 3, the message is finer.\nHowever, it remains quite coarse compared to full revelation, which would be the 45° line, but which is not a Nash equilibrium.\nWith a higher N, and a finer message, the blue area is more important.\nThis implies higher utility.\nDisclosing more information benefits both parties.\n# [Figure 2: Message and utilities for conflict of interest b = 1/20, for N=1, 2, and 3 | upload.wikimedia.org/wikipedia/commons/9/91/Crawford///Sobel.png]","revisionCount":0},"hasMore":false,"nextCursor":null}

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