Ingame theory,cheap talk is communication between players that does not directly affect the payoffs of the game. Providing and receiving information is free. This is in contrast tosignalling, in which sending certain messages may be costly for the sender depending on the state of the world.

This basic setting set byVincent Crawford andJoel Sobel^[1] has given rise to a variety of variants.

To give a formal definition, cheap talk is communication that is:^[2]

1. costless to transmit and receive
2. non-binding (i.e. does not limit strategic choices by either party)
3. unverifiable (i.e. cannot be verified by a third party like a court)

Therefore, an agent engaging in cheap talk could lie with impunity, but may choose in equilibrium not to do so.

Cheap talk can, in general, be added to any game and has the potential to enhance the set of possible equilibrium outcomes. For example, one can add a round of cheap talk in the beginning of the gameBattle of the Sexes, a game in which each player announces whether they intend to go to the football game, or the opera. Because the Battle of the Sexes is acoordination game, this initial round of communication may enable the players to select among multiple equilibria, thereby achieving higher payoffs than in the uncoordinated case. The messages and strategies which yield this outcome are symmetric for each player. One strategy is to announce opera or football with even probability; another is to listen to whether a person announces opera (or football), and on hearing this message say opera (or football) as well.^[3] If both players announce different options, then no coordination is achieved. In the case of only one player messaging, this could also give that player a first-mover advantage.

It is not guaranteed, however, that cheap talk will have an effect on equilibrium payoffs. Another game, thePrisoner's Dilemma, is a game whose only equilibrium is in dominant strategies. Any pre-play cheap talk will be ignored and players will play their dominant strategies (Defect, Defect) regardless of the messages sent.

It has been commonly argued that cheap talk will have no effect on the underlying structure of the game. Inbiology authors have often argued that costly signalling best explains signalling between animals (seeHandicap principle,Signalling theory). This general belief has been receiving some challenges (see work by Carl Bergstrom^[4] andBrian Skyrms 2002, 2004). In particular, several models usingevolutionary game theory indicate that cheap talk can have effects on the evolutionary dynamics of particular games.

In the basic form of the game, there are two players communicating, one senderS and one receiverR.

SenderS gets knowledge of the state of the world or of his "type"t. ReceiverR does not knowt ; he has only ex-ante beliefs about it, and relies on a message fromS to possibly improve the accuracy of his beliefs.

S decides to send messagem. Messagem may disclose full information, but it may also give limited, blurred information: it will typically say "The state of the world is betweent₁ andt₂". It may give no information at all.

The form of the message does not matter, as long as there is mutual understanding, common interpretation. It 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 betweent₁ andt₂".

ReceiverR receives messagem.R updates his beliefs about the state of the world given new information that he might get, usingBayes's rule.R decides to take actiona. This action impacts both his own utility and the sender's utility.

The decision ofS regarding the content ofm is based on maximizing his utility, given what he expectsR to do. Utility is a way to quantify satisfaction or wishes. It can be financial profits, or non-financial satisfaction—for instance the extent to which the environment is protected.→ Quadratic utilities:The respective utilities ofS andR can be specified by the following:$${\displaystyle U^S(a,t)=-(a-t-b{)}^2}$$$${\displaystyle U^R(a,t)=-(a-t{)}^2}$$

The theory applies to more general forms of utility, but quadratic preferences makes exposition easier. ThusS andR have different objectives ifb ≠ 0. Parameterb is interpreted asconflict of interest between the two players, or alternatively as bias.U^R is maximized whena = t, meaning that the receiver wants to take action that matches the state of the world, which he does not know in general.U^S is maximized whena = t + b, meaning thatS wants a slightly higher action to be taken, ifb > 0. SinceS does not control action,S must obtain the desired action by choosing what information to reveal. Each player's utility depends on the state of the world and on both players' decisions that eventually lead to actiona.

We look for an equilibrium where each player decides optimally, assuming that the other player also decides optimally. Players are rational, althoughR has only limited information. Expectations get realized, and there is no incentive to deviate from this situation.

Crawford and Sobel characterize possibleNash equilibria.

• There are typicallymultiple equilibria, but in a finite number.
• Separating, which means full information revelation, is not a Nash equilibrium.
• Babbling, which means no information transmitted, is always an equilibrium outcome.

When interests are aligned, then information is fully disclosed. When conflict of interest is very large, all information is kept hidden. These are extreme cases. The 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.

More generally:

• There existsN^* > 0 such that for allN with1 ≤ N ≤ N^*,
• there exists at least an equilibrium in which the set of induced actions has cardinalityN; and moreover
• there is no equilibrium that induces more thanN^* actions.

While messages could ex-ante assume an infinite number of possible valuesμ(t) for the infinite number of possible states of the worldt, actually they may take only a finite number of values(m₁, m₂, . . . , m_N).

Thus an equilibrium may be characterized by a partition(t₀(N), t₁(N). . . t_N(N)) of the set of types [0, 1],where0 = t₀(N) < t₁(N) < . . . < t_N(N) = 1. This partition is shown on the top right segment of Figure 1.

Thet_i(N)'s are the bounds of intervals where the messages are constant: fort_i-1(N) < t < t_i(N), μ(t) = m_i.

Since actions are functions of messages, actions are also constant over these intervals:fort_i-1(N) < t < t_i(N),α(t) = α(m_i) = a_i.

The action function is now indirectly characterized by the fact that each valuea_i optimizes return for theR, knowing thatt is betweent₁ andt₂. Mathematically (assuming thatt is uniformly distributed over [0, 1]),

${\displaystyle a_i=\overline{a}(t_{i-1},t_i)=\mathrm{a}\mathrm{r}\mathrm{g}\underset{a}{max}{\int }_{t_{i-1}}^{t_i}{U}^R(a,t)dt}$

→Quadratic utilities:

Given thatR knows thatt is betweent_i-1 andt_i, and in the special case quadratic utility whereR wants actiona to be as close tot as possible, we can show that quite intuitively the optimal action is the middle of the interval:$${\displaystyle a_i=\frac{t_{i-1}+t_i}{2}}$$

Att = t_i, The sender has to be indifferent between sending either messagem_i-1 orm_i.${\displaystyle U^S(a_i,t_i)=U^S(a_{i+1},t_i)}$      1 ≤ i≤ N-1

This gives information aboutN and thet_i.

→ Practically:We consider a partition of sizeN.One can show that$${\displaystyle t_i=t_{1}i+2bi(i-1)t_1=\frac{1-2bN(N-1)}{N}}$$

N must be small enough so that the numerator is positive. This determines the maximum allowed value$${\displaystyle N^{\ast }=⟨-\frac{1}{2}+\frac{1}{2}\sqrt{1+\frac{2}{b}}⟩}$$ where${\displaystyle ⟨Z⟩}$ is the ceiling of${\displaystyle Z}$, i.e. the smallest positive integer greater or equal to${\displaystyle Z}$. Example: We assume thatb = 1/20. ThenN^* = 3. We now describe all the equilibria forN=1,2, or3 (see Figure 2).

N = 1: This is the babbling equilibrium.t₀ = 0, t₁ = 1;a₁ = 1/2 = 0.5.

N = 2:t₀ = 0, t₁ = 2/5 = 0.4, t₂ = 1;a₁ = 1/5 = 0.2, a₂ = 7/10 = 0.7.

N = N^* = 3:t₀ = 0, t₁ = 2/15, t₂ = 7/15, t₃ = 1;a₁ = 1/15, a₂ = 3/10 = 0.3, a₃ = 11/15.

WithN = 1, we get thecoarsest possible message, which does not give any information. So everything is red on the top left panel. WithN = 3, the message isfiner. However, it remains quite coarse compared to full revelation, which would be the 45° line, but which is not a Nash equilibrium.

With a higherN, and a finer message, the blue area is more important. This implies higher utility. Disclosing more information benefits both parties.

• Costly signaling theory in evolutionary psychology
• Game theory
• Handicap principle
• Screening game
• Signaling game

• Crawford, V. P.; Sobel, J. (1982). "Strategic Information Transmission".Econometrica.50 (6):1431–1451.CiteSeerX 10.1.1.461.9770.doi:10.2307/1913390.JSTOR 1913390.
• Farrell, J.;Rabin, M. (1996)."Cheap Talk".Journal of Economic Perspectives.10 (3):103–118.doi:10.1257/jep.10.3.103.JSTOR 2138522.
• Robson, A. J. (1990)."Efficiency in Evolutionary Games: Darwin, Nash, and the Secret Handshake"(PDF).Journal of Theoretical Biology.144 (3):379–396.Bibcode:1990JThBi.144..379R.doi:10.1016/S0022-5193(05)80082-7.PMID 2395377.
• Skyrms, B. (2002)."Signals, Evolution and the Explanatory Power of Transient Information"(PDF).Philosophy of Science.69 (3):407–428.doi:10.1086/342451.S2CID 15843361.
• Skyrms, B. (2004).The Stag Hunt and the Evolution of Social Structure. New York: Cambridge University Press.ISBN 0-521-82651-9.

• Game theory
• Asymmetric information

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