Games Mathematicians Play

The word "game" appears to be very ancient. It is a descendent of the Old High Gothic word "gomen", which means fellowship, and is thus related to gamble, gambol, and hence to gambit. It looks sort of like the Greek word "agon" (the ancestor of "agony": the Greeks took competition VERY seriously), and rather like the Sanskrit word "gam", which seems to refer more to large assemblies of people, animals, gods, etc., and hence to ... counting. My guess is that "game" is an Indo-European word. This raises the question of where the Arabic word "layba" came from: perhaps it is African (the Ethiopians, i.e., the Abyssinians, were in invasion mode way back when).
Language can tell us something about how people look at notions. The Romans were more decadent than the Greeks, and the Latin word "ludus" is more fun-oriented than "agon" (and more decadence-oriented than "gomen"). I have no idea where the Romans got "ludus", but it does live on, e.g., on the `play on words', i.e., "joke". Speaking of "play", the Greeks did not think much of it (from "paidos", or child, we get "paizo", or play, apparently not anything so serious as games). The Romans used the word "ludere"??, and the Arabs "yalayb": perhaps they knew something we don't.
It is not clear when mathematicians got interested in games. They probably always were, but there was a problem. Mathematics is serious and it is improper to approach such matters with an inappropriate levity. Fortunately, greed will find a way.
The classical tale of the origin of probability is that the Chevalier de Mere, a jaded roue, wanted to know how to adjucate a game. He went to M. Mersenne, who maintained a sort of pre-electronic newsgroup, and he passed the problem on to two amateurs, a customs official (Fermat) and a theologian (Pascal). For a no doubt accurate account of this tale, click here. Anyway, since then, dice, cards, and other such devices have found their way into studying a subject whose real objects of study were physics and finance.
(Here is an unpleasant truth behind this tale of innocent sin. During the High Middle Ages, when Europeans started getting filthy rich again, they discovered The Love of Money (see I Timothy 6:10). During the Renaissance --- the Late High Middle Ages --- Europeans went out on sea voyages to trade in nutmeg, gold, pepper, porcelain, sugar, human flesh, and so on. The profit margins were up to 1000 %, but the there were risks, like storms and pirates.
(This led to two inventions. One was insurance, which is an obvious application of probability. The other was an updated version of International Law, that said that when a privateer from one country pirated the cargo of a ship of another country, the lawyers should battle the matter out in a neutral prize court. Those cynical lawyers were soon interested in things like the odds of winning. The connection between probability and law still stands: S. D. Poisson wrote a book on the probabilities of correct decisions in criminal cases: it is a commentary on our state of denial that after two centuries, this book by a major mathematician has yet to be translated into English.