An old book we found

One of the things we found recently when clearing up at Melissa’s father’s Bridge club was the 1954 title Theory of Games and Statistical Decisions by David Blackwell and M.A. Girshick.

It has some interesting definitions about games.

“A game is characterized by a set of rules having a certain formal structure, governing the behaviour of certain individuals or groups, the players.”

“Broadly speaking, the rules provide that the game shall consist of a finite sequence of moves in a specified order and the nature of each move is prescribed. Moves are one of two kinds, personal moves and chance moves. A personal move is a choice by one of the players of one of the specified, possibly infinite, set of alternatives; for instance, each move in chess is a personal move; the first move is a choice by White of 1 of 20 specified alternatives. The actual decision made in a particular play of a game at a given personal move is called the choice at that move. A chance move also results in the choice of one of a specified set of alternatives; here the alternative is selected not by one of the players, but by a chance mechanism, with the probabilities with which the mechanism selects the various alternatives specified by the rules of the game.”

“The rules specify, as a function of the choices and outcomes at the successive moves, when the game shall terminate and the score, not necessarily numerical, that is to be assigned to each player.”

The example games given are chess and bridge and there are various coin tossing games devised for the statistical work. It was a probably a fair selection of games for the intended audience back in the early fifties.

The “finite sequence of moves” part of the definition probably caters for the difference between games like chess and bridge where a player’s turn consists of a sequence of one move and that of many games today where a player’s turn may consist of a number of moves or actions.

In a wargame a player's turn generally consists of a number of personal moves, usually moving a unit and/or instigating combat. In the case of moving a unit the outcome is known upfront (to excuse the pun). However, if the move is, or includes, instigating combat then a chance move is often required to determine the resolution of the combat and thus the outcome of the move. This may be the rolling of a die or the playing of cards. It could be argued that your playing of a card is a personal move, but if you do not know what card your opponent is going to play, or your orbital mind control lasers are down for maintenance, then the overall resolution falls more into the category of chance.

In many games a chance mechanic has determined the set of alternatives that you may choose from, e.g. the deal in a standard trick taking game, the plantation draw in Puerto Rico or the face up cards that available for selection in Ticket to Ride or Around the World in 80 Days. You are making a personal move, but your selection is restricted by the game’s prior chance move.

In auction games I see bidding as a personal move. You are choosing what to bid based on what you think the item being auctioned is worth, both to yourself and to other players. There may well be imperfect information involved, e.g. do you know how much money the other players actually have, or is your estimate of the item’s worth accurate? I find, at least in the first few plays of a game, my estimate is often incorrect, or at least at odds to that of other player’s estimates.

Your standard draw a card, play a card game is a sequence of chance move then personal move. Corollary play a card, draw a card would be personal move, chance move.

A roll and move game like Snakes & Ladders is chance move game. Once you have your number there is only one alternative available to you. In a game like Formula Dé there is also the chance move with the roll of the die, but there is a personal aspect both in the choosing of which gear you are in and thus which die is rolled and exactly where the car will move as there is usually more than one alternative as to where to move the car.

The mathematics involved in the statistics in this book was enough to give me a nose bleed, it is aimed at “graduate students” in statistics, or post graduates as we would call them in Australia. For “an excellent treatment of numerous aspects of game theory” the authors recommend J.C.C. McKinsey’s Introduction to the Theory of Games, McGraw-Hill, 1952.

Possibly there may be some more recent and slightly more accessible to the layperson books around too :-)

Mmm meeples taste like…