Message Exchange Games in Strategic Contexts

Abstract

When two people engage in a conversation, knowingly or unknowingly, they are playing a game. Players of such games have diverse objectives, or winning conditions: an applicant trying to convince her potential employer of her eligibility over that of a competitor, a prosecutor trying to convict a defendant, a politician trying to convince an electorate in a political debate, and so on. We argue that infinitary games offer a natural model for many structural characteristics of such conversations. We call such games message exchange games, and we compare them to existing game theoretic frameworks used in linguistics—for example, signaling games—and show that message exchange games are needed to handle non-cooperative conversation. In this paper, we concentrate on conversational games where players’ interests are opposed. We provide a taxonomy of conversations based on their winning conditions, and we investigate some essential features of winning conditions like consistency and what we call rhetorical cooperativity. We show that these features make our games decomposition sensitive, a property we define formally in the paper. We show that this property has far-reaching implications for the existence of winning strategies and their complexity. There is a class of winning conditions (decomposition invariant winning conditions) for which message exchange games are equivalent to Banach- Mazur games, which have been extensively studied and enjoy nice topological results. But decomposition sensitive goals are much more the norm and much more interesting linguistically and philosophically.

Appendix: Proof of Proposition 1

Proof

Let s(⋅|m) be a receiver strategy. Let μ(⋅|m) be a probability distribution over sender types such that s is rational given belief in μ and such that $s(a^{*}_{t_{\textsc {good}},m}|m) > 0$. By definition, if s is rational given μ, $a^{*}_{{t_{\textsc {good}}},m}$ must be a best response to m. In particular $a^{*}_{{t_{\textsc {good}}},m}$ must yield a better (or equal) expected utility that $a^{*}_{{t_{\text {bad}}},m}$ which can be written as:

$${\sum}_{t \in T} \mu(t|m)u_{R}(t,m,a^{*}_{t_{\textsc{good}},m}) - {\sum}_{t} \mu(t|m)u_{R}(t,m,a^{*}_{t_{\textsc{bad}},m}) \ge 0 $$

that is to say

$$\left(\begin{array}{l} \sum\limits_{t \in T_{\textsc{good}}} \mu(t|m)\left((u_{R}(t,m,a^{*}_{t_{\textsc{good}},m}) - u_{R}(t,m,a^{*}_{t_{\textsc{bad}},m}) \right) \\ -\sum\limits_{t \in t_{\textsc{bad}}} \mu(t|m) \left(u_{R}(t,m,a^{*}_{t_{\textsc{bad}},m}) - u_{R}(t,m,a^{*}_{t_{\textsc{GOOD}},m}) \right) \end{array} \right) \ge 0 $$

Notice that both terms of the above difference are positive (the first term on the left is strictly positive since t _good∈T _good). Let

$$\begin{array}{l}\delta_{\textsc{good}} = max_{t \in T_{\textsc{good}}} (u_{R}(t,m,a^{*}_{t_{\textsc{good}},m}) - u_{R}(t,m,a^{*}_{t_{\textsc{bad}},m})) \textnormal{ and}\\ \delta_{t_{\textsc{bad}}} = u_{R}(t_{\textsc{bad}},m,a^{*}_{t_{\textsc{bad}},m}) - u_{R}(t_{\textsc{bad}},m,a^{*}_{t_{\textsc{good}},m}). \end{array}$$

Since t _bad∈T _bad we have:

$${\sum}_{t \in T_{\textsc{good}}} \mu(t|m)\delta_{\textsc{good}} - \mu(t_{\textsc{bad}}|m)\delta_{t_{\textsc{bad}}} \ge \left(\begin{array}{l} \sum\limits_{t \in t_{\textsc{good}}} \mu(t|m)\left((u_{R}(t,m,a^{*}_{t_{\textsc{good}},m}) - u_{R}(t,m,a^{*}_{t_{\textsc{bad}},m}) \right) \\ -\sum\limits_{t \in t_{\textsc{bad}}} \mu(t|m) \left(u_{R}(t,m,a^{*}_{t_{\textsc{bad}},m}) - u_{R}(t,m,a^{*}_{t_{\textsc{goog}},m}) \right) \end{array} \right) $$

Hence we must have

$${\sum}_{t \in T_{\textsc{good}}} \mu(t|m)\delta_{\textsc{good}} - \mu(t_{\textsc{bad}}|m)\delta_{t_{\textsc{bad}}} = \mu(T_{\textsc{good}}|m)\delta_{\textsc{good}} - \mu(t_{\textsc{bad}}|m)\delta_{t_{\textsc{bad}}} \ge 0 $$

and since $\delta _{t_{\textsc {bad}}} > 0$ by hypothesis we have $\mu (t_{\textsc {good}}|m)\frac {\delta _{t_{\textsc {good}}}}{\delta _{t_{\textsc {bad}}}} \ge \mu (t_{\textsc {bad}} |m)$ which concludes the proof. □