## Translating games

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Can you translate games, in the sense that you can translate languages? More precisely, can you translate an instance of one game — a match or a round or whatever — into an instance of another game, as you can translate a sentence or a paragraph of Chinese into a sentence or a paragraph of English?

Helen DeWitt sensibly says that you can't. But I think that there's more to the story.

Here's Helen's argument:

Suppose I grow up in a family where people obsessively play Hearts. We switch around between different versions of the game – sometimes we play Black Maria, where the Queen of Spades costs you 13 points and you pass on three cards to the left before you begin play, sometimes we invent twists of our own. I also have four friends: A lives in a family of chess fanatics, B lives in a family of bridge fanatics, C lives in a family of go fanatics, D lives in a family of poker fanatics.

What I see at once is something remarkable. Languages are translatable, more or less; it may be more or less tricky, but it's intelligible to speak of Chinese being translated into Turkish AND Arabic AND English. Games are not translatable. Chess is a game for two players with complete information; you can't "explain" what's going on in a chess game in terms of bridge, which is a game for two sets of partners with imperfect information, a mixture of skill and chance which depends on skilful sharing of information between partners. And you can't "explain" either in terms of poker, which is a game for an indeterminate number of players, a mixture of skill and chance in which sharing of information between players would in fact be collusion and outlawed. A game is intelligible on its own terms – which means, paradoxically, that you can play a game with someone whose language you don't know, provided you both know the rules of the game.

This is certainly true about hearts, chess, and poker. But there are some special games, probably never played by any family for fun, where it may be false. According to the Stanford Encyclopedia of Philosophy's entry on Logic and Games,

Important examples are semantic games used to define truth, back-and-forth games used to compare structures, and dialogue games to express (and perhaps explain) formal proofs.

In this realm, there are circumstances that make it plausible to talk about a translation from one game to another. For example, suppose we provide game-theoretic semantics for two different logics that express propositions about the same sorts of worlds, or two different programming languages that apply similar sets of actions to the same sorts of states. Then it may make sense to talk about translating "games" of one kind into "games" of another kind, just as it makes sense to talk about translating logical expressions or translating computer programs.

All sorts of relationships are possible here, I think. Two game-systems might be isomorphic, in the sense that every game-instance in game-sytem A is translatable into a game-instance in system B, and vice versa. Or game A might properly include game B, in the sense that every game-instance of B can be translated into A, but not vice versa; or two games might have both an intersection and a symmetric difference that are non-null.

This is irrelevant to Helen's point about the impossibility of radical translation between chess and hearts. But she was responding to a post where Language Hat quoted Remy de Gourmont on the need for writers to break the rules of their cultural tradition ("Le crime capital pour un écrivain c'est le conformisme, l'imitativité, la soumission aux règles et aux enseignements"). And her next two sentences were

You don't understand a game in terms of some other game, you understand it by learning to play it – but the more games you play, the more you will understand about the radical otherness of games. And this, it seems to me, is the sense in which Gourmont thinks each writer will develop his own aesthetics.

Perhaps the cultural patterns in question are more like logic or computer programming than like chess or hearts, in the sense that they're not just systems for scoring a sequence of ritualized actions, but also ways to express, communicate, or modify meanings.

In this connection, it may be interesting to raise a question that puzzles game-theoretic logicians. Quoting again from the Stanford Encyclopedia article:

In most applications of logical games, the central notion is that of a winning strategy for the player ∃. Often these strategies (or their existence) turn out to be equivalent to something of logical importance that could have been defined without using games — for example a proof. But games are felt to give a better definition because they quite literally supply some motivation: ∃ is trying to win. This raises a question that is not of much interest mathematically, but it should concern philosophers who use logical games. If we want ∃'s motivation in a game G to have any explanatory value, then we need to understand what is achieved if ∃ does win. In particular we should be able to tell a realistic story of a situation in which some agent called ∃ is trying to do something intelligible, and doing it is the same thing as winning in the game. As Richard Dawkins said, raising the corresponding question for the evolutionary games of Maynard Smith,

The whole purpose of our search … is to discover a suitable actor to play the leading role in our metaphors of purpose. We … want to say, ‘It is for the good of … ‘. Our quest in this chapter is for the right way to complete that sentence. (The Extended Phenotype, Oxford University Press, Oxford 1982, page 91.)

For future reference let us call this the Dawkins question. In many kinds of logical game it turns out to be distinctly harder to answer than the pioneers of these games realised.

(A notational note: the two players in these games are conventionally known as ∃ and ∀, which are the symbols for the existential and universal quantifiers, but here are pronounced "Eloise" and "Abelard".)

In reference to Jaakko Hintikka's influential game-theoretic semantics for the predicate calculus, the article adds:

Computer implementations of these games of Hintikka proved to be a very effective way of teaching the meanings of first-order sentences. One such package was designed by Jon Barwise and John Etchemendy at Stanford, called ‘Tarski's World’. Independently another team at the University of Omsk constructed a Russian version for use at schools for gifted children.

In the published version of his John Locke lectures at Oxford, Hintikka in 1973 raised the Dawkins question (see above) for these games. His answer was that one should look to Wittgenstein's language games, and the language games for understanding quantifiers are those which revolve around seeking and finding. In the corresponding logical games one should think of ∃ as Myself and ∀ as a hostile Nature who can never be relied on to present the object I want; so to be sure of finding it, I need a winning strategy. This story was never very convincing; the motivation of Nature is irrelevant, and nothing in the logical game corresponds to seeking. In retrospect it is a little disappointing that nobody took the trouble to look for a better story.

[Update: Robert Furber writes:

Translating between games can in some cases be defined.

Andre Joyal defined a category (in the sense of category theory) of games based on John Conway's theory of two-player games with complete information. A translation into English of the original note is here.

Because Conway's theory allows games to be added and negated, I can think of a strategy for A – B to be a thing that would be a strategy for A if only I had a strategy for B (the article explains this in more detail). This is rather like two things:

1. The way implication works under the Brouwer-Heyting-Kolmogorov interpretation of (intuitionistic) logic. A proof of A -> B is a function that turns a proof of A into a proof of B.

2. Subtraction of vectors. A – B tells you how to get from B to A. This relates to translation in the sense it's used in geometry.

]

[Update #2: Vykromond Jorilliam points to a very different sort of idea —

Raph Koster's attempts at establishing a "game grammar" (http://www.gamasutra.com/view/feature/1979/defining_games_raph_kosters_game_.php) might be relevant to this topic as an approach to attempt to identify a formal "language" for games that would allow for, among other things, translation.

This is an interesting-sounding idea, reminiscent in some ways of <a href="http://nickm.com/">Nick Montfort</a>'s work on interactive fiction. But I can't tell from the cited interview, or from any of the links it contains, what Koster actually means, and whether there's more to his idea so far than a perceived need and a general prescription for the shape of a solution. ]