page_title: Cheap talk
section_title: Theorem
hierarchical_section_title: Cheap talk / Crawford and Sobel's original article / Theorem
caption_attribution_description: English: Cheap talk communication setting
context_page_description: 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 is of an expert trying to explain the state of the world to an uninformed decision maker. 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 Crawford and Sobel has given rise to a variety of variants.
  To give a formal definition, cheap talk is communication that is:
  costless to transmit and receive
  non-binding
  unverifiable
  Therefore, an agent engaging in cheap talk could lie with impunity, but may choose in equilibrium not to do so.
context_section_description: 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, . . . , mN).
  Thus an equilibrium may be characterized by a partition (t₀(N), t₁(N). . . tN(N)) of the set of types [0, 1],
  where 0 = t₀(N) < t₁(N) < . . . < tN(N) = 1. This partition is shown on the top right segment of Figure 1.
  The tᵢ(N)’s are the bounds of intervals where the messages are constant: for tᵢ₋₁(N) < t < tᵢ(N), µ(t) = mᵢ.
  Actions.
  Since actions are functions of messages, actions are also constant over these intervals:
  for tᵢ₋₁(N) < t < tᵢ(N), α(t) = α(mᵢ) = aᵢ.
  The 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. Mathematically (assuming that t is uniformly distributed over [0, 1]),
  → Quadratic utilities:
  Given 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:
  Indifference condition.
  What happens at t = tᵢ? The sender has to be indifferent between sending either message mᵢ₋₁ or mᵢ.
         1 ≤ i≤ N-1
  This gives information about N and the tᵢ.
  → Practically:
  We consider a partition of size N.
  One can show that 
  N must be small enough so that the numerator is positive. This determines the maximum allowed value 
   where  is the ceiling of , i.e. the smallest positive integer greater or equal to . 
  
  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.
caption: Figure 1: Cheap talk communication setting
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