**4. Model**

Consider a game **G** with *n* players and finite action spaces *Si* for *i* = 1, ..., *n*. <sup>13</sup> Payoffs are given by *ui* for *i* = 1, ..., *n*. It will be simplest to think of **G** in normal form. **G** is played exactly once, though **G** itself may be a repeated game. Before this happens, there is a **conversation** C(**G**), defined as follows. Each player begins the pregame conversation with a totally mixed prior **forecast** *π<sup>i</sup>* = *π*<sup>1</sup> *<sup>i</sup>* ∈ Δ(*Si*) about his or her behavior. The forecasts are common knowledge among all the players. At each

<sup>12</sup> Naturally, since full rationality is assumed, we could endlessly iterate the process, but there is no need.

<sup>13</sup> The assumption of finiteness can be weakened.

round *t* = 1, 2, 3... of the conversation, player *i* announces *m<sup>t</sup> <sup>i</sup>* ∈ *Si*. The announcements are made simultaneously by all players in each round.14

Let *NE*(*G*) <sup>⊆</sup> *<sup>n</sup>* × *i*=1 Δ(*Si*) be the set of Nash equilibria of **G**, and define *Ei* ⊆ *Si* by:

$$E\_i = \{ s\_i \in \mathcal{S}\_i \mid \exists \sigma \in NE(G) \text{ with } s\_i \in supp(\sigma\_i) \}\ .$$

This set constitutes the self-committing actions for player *i*. At *t* = 1, any *m*<sup>1</sup> *<sup>i</sup>* ∈ *Ei* is said to be **credible**. If *m*<sup>1</sup> *<sup>i</sup>* was credible, then we define:

$$
\pi\_i^2 = (T\pi\_i^1 + m\_i^1)/(T+1).
$$

for some fixed *T*, which can be chosen to be large relative to the scale of the strategy space and payoffs in the underlying game. This captures the slow updating process of prior forecasts by credible announcements. In a similar fashion, the **appearance** is given by *p*<sup>2</sup> *<sup>i</sup>* = *<sup>m</sup>*<sup>1</sup> *<sup>i</sup>* . If the initial announcement was not credible, then the forecast is not updated, and the appearance is undefined. Recursively, we now define *m<sup>t</sup> <sup>i</sup>* to be credible when:

$$m\_i^t \in \mathfrak{e}BR\_i(\underset{j\neq i}{\times} q\_j^t) \text{ with } q\_j^t = \pi\_j^t \text{ or } p\_j^t \forall j.$$

where *εBRi*(*σ*−1) denotes:

$$\left\{ s\_i \in \mathbb{S}\_i \mid \max\_{s'\_i \in \mathcal{S}\_i} u\_i(s'\_{i'} \sigma\_{-1}) - u\_i(s\_{i'} \sigma\_{-i}) < \varepsilon \right\}$$

for some arbitrarily small *ε* > 0. If *m<sup>t</sup> <sup>i</sup>* is not credible,<sup>15</sup> then *<sup>π</sup>t*+<sup>1</sup> *<sup>i</sup>* = *<sup>π</sup><sup>t</sup> <sup>i</sup>* and *<sup>p</sup>t*+<sup>1</sup> *<sup>i</sup>* = *<sup>p</sup><sup>t</sup> i* . If *m<sup>t</sup> <sup>i</sup>* is credible, then we define:

$$\pi\_i^{t+1} = ((T+t-1)\pi\_i^1 + m\_i^1)/(T+t) \text{ and } p\_i^{t+1} = ((t-1)p\_i^t + m\_i^t)/t.$$

Say that player *i*'s appearance *converges* if player *i* never entirely stops making credible announcements and if lim *<sup>t</sup>*→∞*p<sup>t</sup> <sup>i</sup>* exists. If this happens, it is clear that lim *<sup>t</sup>*→∞*π<sup>t</sup> <sup>i</sup>* also exists and is the same; call it *bi* for the belief about player *i*. If the limit exists for all players, then the conversation converges. In this case, we assume that beliefs after the conversation and entering **G** are given by *μ<sup>i</sup>* = × *j*=*i bj*.

**Definition 1.** *An acceptable equilibrium (of <sup>G</sup>) is a profile <sup>σ</sup>* <sup>∈</sup> *<sup>n</sup>* × *i*=1 Δ(*Si*) *such that σ* = *b for some belief vector b resulting from a convergent conversation starting at some prior forecasts π; the set of acceptable equilibria is denoted AccE*(*G*)*.*
