“Real-world language use is flexible; language is used to coordinate our mental lives in a process of constant negotiation.”

Robin Clark wants to convince us that the best way to model linguistic behavior is with game theory. As I began this book, I was thinking about the short-comings of these two fields of study. Linguistics has been dogged by accusations of poor falsifiability. You can have (so the argument goes) many competing models of the same phenomenon without any sort of clear sense as to what sort of empirical evidence would decide between these models. And game theory, as a means of modelling human behavior, certainly has its critics as well. There are games with identical strategic structures that people will play very differently. There are games where people simply will not land on a Nash equilibrium (even simple games when bounded rationality isn’t an issue), and the only way to correct the model is by a possibly ad-hoc adjustment of the utility functions. So when game theory and linguistics meet, will they exacerbate or ameliorate their respective problems? At the end of the book, I’m still not sure, but I do feel optimistic about the situation.

Clark assumes surprisingly little prior understanding, while still remaining interesting to someone who’s read around a bit in these areas. The first three chapters are a discussion of the dominant perspective on mind and meaning in the second half of the twentieth century (functionalism, though somehow that term never comes up); Clark argues that the social nature of meaning makes that philosophy impossible. I didn’t need any convincing here, though I do think I benefited from his particular articulation of the issues. Chapters four and five are overviews of game theory and game theoretic semantics, respectively. Chapter six a discussion of common knowledge. But the real meat of the book comes in the last three chapters, where the focus is on modelling linguistic behavior with games. In chapters seven and eight, people are the players and conversations are the games. In chapter nine, words become the players and meaning is the game. As words aren’t rational agents, but better thought of as randomly deviating populations, we appropriately switch from looking for Nash equilibria to looking for evolutionarily stable strategies.

Science works by looking at the behavior of systems and trying to figure out what that behavior is minimizing/maximizing; science is the backward induction of optimization problems. The innovation of game theory is dropping the assumption that the system is a single unit optimizing a single function; there may be multiple agents each trying to maximize a distinct utility function. It seems, however, that for any given game, you should be able to map to an equivalent system of the first type, with a single function being optimized. For example, suppose we have a generic two-player, two-strategy game that has a unique mixed-strategy Nash equilibrium $(p_1, q_1)$, with player one playing row and player two playing column:

 $q_1$ $q_2(= 1-q_1)$ $p_1$ $(a_1, a_2)$ $(b_1, b_2)$ $p_2$ $(c_1, c_2)$ $(d_1, d_2)$

Then player 1 maximizes her utility by taking the derivative with respect to $q_1$ of the equation $p_1\cdot q_1\cdot a_2 + p_2\cdot q_1\cdot c_2 + p_1\cdot q_2\cdot b_2 + p_2\cdot q_2\cdot d_2$ and setting it equal to zero. Player 2 acts analogously.

But we could suppose there is a single player, whose set of strategies is the four strategies in the cartesian product of two sets of strategies of players 1 and 2 above. For this system to be equivalent to the two player game, we would want the probabilities of these four choices to be $r_1=p_1\cdot q_1$, $r_2=p_1\cdot q_2$, $r_3=p_2\cdot q_1$, $r_4=p_2\cdot q_2$. One should be able to construct a a function of the variables $r_i$ with constants $a_i, b_i, c_i, d_i$ whose maximum takes those values.

The point is that what is being modeled in solving a formal game is not different from what could otherwise be modeled. We want to adopt game theory just in those cases when we believe there are distinct players acting autonomously – and what is counts as an autonomous player if not a human being. Thus it is very natural to think of language as a game. But we are still computing what could otherwise be computed; it still comes down to symbolic manipulation. One might have perfectly good reasons to exile oneself from platonism, but the introduction of game theory to linguistic analysis is, from a phenomenological perspective, a pragmatic move either way.

Ever since I first read Wittgenstein’s Philosophical Investigations, I was convinced that meaning is use. So the primary question for me, than, is how far will game theory take us in that direction? Formal games differ in a fundamental way from his language games. For that giant of twentieth century philosophy, the rules of the game are determined by playing the game. Standard game theory has no mechanism for this sort of thing. I believe this is a major problem. If the social games we play build language, it’s equally true that language builds the social games we play. Language is a game; a game is a language.

There are three ways language might affect the parameters of games: By influencing the utility of outcomes; by changing how we decide our best strategy; by generating our possible strategies. I’ll briefly discuss how this might happen in each case.

Utility functions: We use language to understand the games we’re playing, and what language we use could affect the utility for a given outcome. If I’m playing a prisoner’s dilemma, I might think of “defecting” as “snitching” and “cooperating” as “being loyal to my partner.” My ability to describe my actions with those words might imbue the associated outcomes with increased/decreased utility.

The determination of optimal strategies: As mentioned at the beginning of this article, in the laboratory, human behavior often isn’t aligned with the Nash equilibria, and this can’t always be explained away by a systematic correction of the utility function. (See Goeree and Holt, 2001.) The reason for this may be precisely that language is the intermediary between our utility function and our possible strategies; we use language to compute our best bet, and the informal reasonings of natural language might be very different from the formal reasonings in a mathematical model.

The strategies themselves: This is, I think, the most important point I’d like to make. If the strategies of the games we’re using to model language are not themselves generated, then we’ve lost what’s been gained in the past sixty years linguistic inquiry. Games themselves must have a grammar. For example, in standard game theory, the introduction of new information might change the expected utility of a given strategy, but the set of strategies will not change. In an actual conversation between two people, when one speaker introduces new information, this will change the perceived strategies of the second speaker. (For example, speaker one informs speaker two that she just got back from Barcelona, then this will bring to speaker two’s mind thoughts and questions about traveling and the city of Barcelona.)

If you’re interested in language, this is a worthwhile book for both beginners and more advanced readers. I’m glad I read this one, and I’m glad the research discussed is being done. However, the introduction of game theory to linguistic analysis should be seen as only an initial stride toward a scientific understanding of language as a social phenomenon.

Robin Clark is Associate Professor in the Department of Linguistics at the University of Pennsylvania. Meaningful Games is published by the MIT Press.