Cheap talk

In game 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 to signaling in which sending certain messages may be costly for the sender depending on the state of the world.

One actor has information and the other has ability to act. The informed player can choose strategically what to say and what not to say. Things become interesting when the interests of the players are not aligned. The classic example^{[citation needed]} is of an expert (say, an ecologist) trying to explain the state of the world to an uninformed decision maker (say, a politician voting on a deforestation bill). The decision maker, after hearing the report from the expert, must then make a decision which affects the payoffs of both players.

This basic setting set by Vincent Crawford and Joel 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.

Applications

Game theory

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 Battle of the Sexes. Each player announces whether they intend to go to the football game, or the opera. Because the Battle of the Sexes is a 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. The messages and strategies which yield this outcome are symmetric for each player. They 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, 1996). If they both 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.

그러나 값싼 대화가 평형 보상에 영향을 미칠 것이라는 것은 보장되지 않는다. 또 다른 게임인 '죄수의 딜레마'는 평형만이 지배적인 전략에 있는 게임이다. 플레이 전 값싼 대화는 무시되고 플레이어는 보내는 메시지에 상관없이 지배적인 전략(결함, 결함)을 구사하게 된다.

생물학적 응용

싸구려 이야기는 게임의 기본 구조에 영향을 미치지 않을 것이라는 것이 흔히 주장되어 왔다. 생물학에서 저자들은 종종 값비싼 신호 전달이 동물들 사이의 신호 전달을 가장 잘 설명한다고 주장해왔다. 이러한 일반적인 믿음은 몇 가지 도전을 받아왔다(Carl Bergstrom과^[3] Brian Skyrms 2002, 2004년 작품 참조). 특히, 진화 게임 이론을 사용하는 여러 모델들은 값싼 대화가 특정 게임의 진화 역학관계에 영향을 미칠 수 있다는 것을 보여준다.

크로포드와 소벨의 원문

설정

게임의 기본 형태에는 송신자 S와 수신자 R이 각각 1명씩 통신하는 두 명의 플레이어가 있다.

유형. 송신자 S는 세계의 상태나 그의 "형" t에 대한 지식을 얻는다. 수신자 R은 t를 모른다; 그는 그것에 대해 이전의 믿음만을 가지고 있을 뿐이고, 그의 믿음의 정확성을 향상시키기 위해 S로부터의 메시지에 의존한다.

메시지. S는 메시지 m을 보내기로 결정한다. 메시지 m은 전체 정보를 공개할 수도 있지만, 제한적이고 흐릿한 정보를 줄 수도 있다: 그것은 전형적으로 "세계의 상태는₁ t와₂ t사이에 있다"라고 말할 것이다. 그것은 전혀 정보를 주지 않을 수도 있다.

상호 이해, 공통의 해석이 있는 한 메시지의 형식은 문제가 되지 않는다. 중앙은행 총재의 총성일 수도 있고, 어떤 언어로든 정치적 연설일 수도 있다. 어떤 형태든 결국 "세계의 상태는 t와₁ t사이에₂ 있다"는 의미로 받아들여진다.

조치. 수신기 R은 메시지 m을 수신한다. R은 베이스의 규칙을 이용하여 얻을 수 있는 새로운 정보를 제공함으로써 세계의 상태에 대한 그의 믿음을 갱신한다. R은 조치 a를 취하기로 결정한다. 이 행동은 자신의 효용과 발신자의 효용 모두에 영향을 미친다.

Utility. The decision of S regarding the content of m is based on maximizing his utility, given what he expects R 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 of S and R 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. Thus S and R have different objectives if b ≠ 0. Parameter b is interpreted as conflict of interest between the two players, or alternatively as bias.

U^R 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. U^S is maximized when a = t + b, meaning that S wants a slightly higher action to be taken. Since S 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 action a.

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

Theorem

Crawford and Sobel characterize possible Nash equilibria.

• There are typically multiple 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 exists N^* > 0 such that for all N with 1 ≤ N ≤ N^*,
• there exists at least an equilibrium in which the set of induced actions has cardinality N; and moreover
• there is no equilibrium that induces more than N^* actions.

Messages. While 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, . . . , 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], where 0 = t₀(N) < t₁(N) < . . . < t_N(N) = 1. This partition is shown on the top right segment of Figure 1.

The t_i(N)’s are the bounds of intervals where the messages are constant: for t_i-1(N) < t < t_i(N), µ(t) = m_i.

Actions. Since actions are functions of messages, actions are also constant over these intervals: for t_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 value a_i optimizes return for the R, knowing that t is between t₁ and t₂. Mathematically (assuming that t is uniformly distributed over [0, 1]),

${\displaystyle a_{i}={\bar {a}}(t_{i-1},t_{i})=\mathrm {arg} \max _{a}\int _{t_{i-1}}^{t_{i}}U^{R}(a,t)dt}$

→ Quadratic utilities:

Given that R knows that t is between t_i-1 and t_i, 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:

$${\displaystyle a_{i}={\frac {t_{i-1}+t_{i}}{2}}}$$

Indifference condition. What happens at t = t_i? The sender has to be indifferent between sending either message m_i-1 or m_i. ${\displaystyle U^{S}(a_{i},t_{i})=U^{S}(a_{i+1},t_{i})}$ 1 ≤ i≤ N-1

This gives information about N and the t_i.

→ Practically:

We consider a partition of size N. One can show that

$${\displaystyle t_{i}=t_{1}i+2bi(i-1)\qquad t_{1}={\frac {1-2bN(N-1)}{N}}}$$

N must be small enough so that the numerator is positive. This determines the maximum allowed value

$${\displaystyle N^{*}=\langle {\frac {1}{2}}+{\frac {1}{2}}{\sqrt {1+{\frac {2}{b}}}}\rangle }$$

where ${\displaystyle \langle Z\rangle }$ is the ceiling of ${\displaystyle Z}$, i.e. the smallest positive integer greater or equal to ${\displaystyle Z}$.

Example: We assume that b = 1/20. Then N^* = 3. We now describe all the equilibria for N=1, 2, or 3 (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.

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

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

See also

• Game theory
• Handicap principle
• Screening game
• Signaling game
• Small talk
• Trash-talk

Notes

References

• 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. 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.
• Skyrms, B. (2004). The Stag Hunt and the Evolution of Social Structure. New York: Cambridge University Press. ISBN 0-521-82651-9.