P[ :C]: Games
Possible probabilities are a generalisation of conventional, precise, probabilities in which one supposes that more than one probability function may be possible. Following Dirac and Good, a set of possible probabilities appropriate to a context, C, is denoted P< :C>. Here the relationship with von Neumann and Morgenstern’s Theory of Games, which they base on conventional precise probabilities, is discussed.
4. Structure of the Theory
Many game settings have multiple stable ‘standards of behaviors’ that are equally good, such as which side of the road everyone drives on. Once these are settled one gets a ‘game’, such as driving to work in a timely and safe manner. We can take each possible set of standards to be a context, which may then give rise to a game that has stable probabilities P< :C>. We may also go beyond the stable situation, to consider bounds on possible transitions. I may illustrate this with some data from the Middle East.
10. Axiomatic Formulation
10.1 The Axioms and their Interpretations
It is assumed that:
10:1:e* The probabilities of the various alternative choices at a chance move … behave like probabilities belonging to disjunct but exhaustive probabilities.
Thus, it is assumed that there is some conventional knowable precise ‘objective’ probability distribution, p( ), so that each P< :C> contains a single probability function. We can generalise this to empirical games, where do not know the precise values of the probability functions, and to muddled behaviours, where the probability functions vary, as when adaptation is taking place.
Similarly, 10:1:c* assumes that there is a definite ‘pattern of choice’ for each player, and 10:1:a* assumes that there is an rule-based umpire, with no discretion. Thus it is assumed that there is a definite context, C, that ‘sets the rules’. This rules out the situation where two people start playing assuming different variants of the rules, where the umpire has some discretion, or where players will need to negotiate rules as they go along. These situations are ‘possible games’, an extension of the concept of possible probabilities. (We could think of these as two-level games, where players can play the ostensible game or game the rules. But the gaming of the rules may not itself have formal rules.)
11 Strategies …
11.1 The Concept of a Strategy …
Imagine now that each player … , instead of making each decision as the necessity for it arises, makes up his mind in advance for all possible contingencies … .
… The only extra burden our assumption puts on the player is the intellectual one to be prepared with a rule of behaviour for all eventualities,- although he is to go through one play only. But this is an innocuous assumption within the confines of a mathematical analysis.
The requirement here is that a player should be able to devise a strategy and then stick to it, come what may. This would only be possible if the player had fixed objectives. Thus, for example, if a player was enjoying stringing his opponent along, or – alternatively – got bored and wanted to end the game, to be ‘rational’ they could not let this affect their play. To accommodate these changes they would need something more flexible, perhaps based on ‘possible strategies’, to be adapted and adopted as the necessity arises. We could perhaps develop the notion of having a strategy for playing imperfectly understood or muddled possible games, with possible probabilities.
11.2 The Final Simplification …
The player’s judgement must be directed solely by [the] “mathematical expectation” … .
For a fixed, settled, game, mathematical expectation has meaning and can ‘in principle’ be estimated. For a less definite situation, such as where the status quo game may become unsettled or their are muddled behaviors, mathematical expectations take sets of values, E< :C> that are not so readily comparable. There is an element of genuine choice.
Using possible probabilities is a genuine extension of the language of conventional game theory, allowing one to at least represent more interesting potential behaviors, even those that challenge the established order.