A proposed formal model for human behaviour, illustrated with an elementary game

The Game of Murder

Simon G. Sheppard
= £10


Procedural Analysis (PA) is a novel psychological approach which borrows on evolutionary psychology, game theory and biological sex differences (see Moir & Jessel, 1989 for a summary). This article will give a few details of the origins of PA and highlight a technical proposal for expressing human games formally.

The original investigations took place in Amsterdam and trials of various kinds were a full-time activity between 1992 and 1994. The focus of the investigations was sexual selection, especially mate selection by females, and how control is achieved. Constant reference was made to evolutionary precepts to maintain consistency and prevent deviation.

Many extremes of behaviour were stark in that sexually charged environment. Feminism was a potent force, disinhibition was encouraged, and sexual enjoyment was officially promoted as healthy and to be pursued. The Dutch encouragement of free expression permitted ready analysis and was in contrast to the traditional British reserve which was the author’s background. The work was often very arduous psychologically.

The scheme is proposed as a theoretical system but nonetheless seems to accord with much observed human behaviour. Here a procedure is a behavioural sequence by which a protagonist advances (or attempts to advance) in their competition with an opponent. A procedure should be capable of being modelled mathematically while a complete human strategy set, because of its extent, is unlikely to be. Notwithstanding this large strategy set, people are more predictable than they like to suppose, in the main. PA is internally consistent and capable of describing basic, readily observable mechanisms.

In the well-known work by Axelrod and Hamilton (1981) was stated that “as one moves up the evolutionary ladder in neural complexity, game-playing behaviour becomes richer.“ Thus we can expect human behaviour to be a rich source of games. Sexual behaviour, the ‘battle of the sexes’ being practically the oldest conflict of all, should provide an abundance of game archetypes. The object is to generalise the revealed mechanisms, since sexual behaviour must be the fount of all behaviour. The analysis is not concerned with individuals particularly but with the antipodal male and female strategies.

As has been argued by Wolpert (1992: 128-135) and others, a good theory is one which simplifies, not complicates, and this is the primary object of the approach. The mathematics of these games is not difficult: the challenge lies in establishing the models. Asymmetric games such as male-female games can be oscillatory over time, that is, cyclic (Smith, 1982: 130-131; 199-202). Some of the game-modelling concepts independently arrived upon below have been anticipated, notably by Parker (1978).

Although the essential model in PA applies a male-female dichotomy, the following attempts to describe a primal human game, the game of murder. First however, is a schema for costs.


A set of ten costs has been proposed (Sheppard, 2002). In this scheme, each cost (benefit) is a component of a ten-dimensional vector. For example, loss of face and feelings of guilt are included in Diminishment of Self, having a negative value, while the satisfaction of seeing an adversary’s demise is Enhancement of Self, when the component has a positive value.

Accordingly each payoff is a 10-tuple. The established notation is of Reward, Sucker’s, Temptation and Punishment payoffs. For one payoff vector R

R = [ r1, r2, ..., r10 ]

Time, another of the ten costs, is a linear and indubitably important component. Conventionally, the expected payoff of a game is the payoff weighted by the probability of receiving the payoff. This may be appropriate in overview, but is inadequate for mid-game appraisals, for decisions made ‘on the fly.’ An obvious possibility is that a protagonist could promote a false anticipation of the opponent’s payoff, with their game decisions being influenced by that false perception. An anticipated payoff can change solely with the passage of time (e.g. waiting for the repayment of a loan; “temporal discounting”).

The perception of time might even be altered to the advantage of a player (e.g. changing a clock, advocating that the only consideration is one’s own lifetime). Is there any variable in a human game which is not susceptible to manipulation?

The Game of Murder

In the following game, which is proposed as a basic model, the Protagonist X wants to kill Y. His Opponent is his whole environment Ω; of course Y ⊂ Ω (Y is contained within Ω). For simplicity we neglect the Opponent’s payoffs, which nevertheless may resemble those of a large fish being serviced by a cleaner: the cleaner fish eats parasites from the larger fish, which reciprocates by not eating its cleaner (Trivers, 1971). In all these games, each player can either Cooperate or Defect.

In contemplating a murder the dominant features of Ω in a Western setting would be the police, judiciary and prison service but this would not be the case, say, in an isolated African village. Another scenario is the Protagonist shooting his life-long enemy from his deathbed, at which point his punishment might be inconsequential.

  1. C/C′ is the situation in which X cooperates by not killing, Ω cooperates by not punishing (Protagonist receives the R payoff).
  2. C/D′ corresponds to the Protagonist suffering the punishment for a murder unjustly (S payoff).
  3. D/C′ occurs if X commits a murder but is not punished, for example if the authorities were non-existent or inactive, say if X were able to successfully disguise his defection (a familiar murder scenario which yields T, the highest payoff of all).
  4. D/D′ occurs if X commits a murder and Ω delivers a punishment (P payoff).

Furthermore each payoff vector has a simple value, comparable to a modulus, expressing its ‘overall benefit.’ (It could be argued that there is no such thing as a conflict of desires, only an uncertain appraisal of the overall benefit.) This scalar value might also be called the ‘mean benefit value’ or ‘overall payoff’ for a player having appraised the various components of the payoff vector. Hence if money was an individual’s main interest, that component would dominate his appraisal of overall benefit.

So T is a ten-tuple and let tX be X’s appraisal of the overall Temptation payoff, a scalar value. Another player will almost certainly have a different appraisal, or different appraisal schemes can be applied, such as Parker’s (1978) “net positive reward.”

Plus of course an appraisal can change as the game progresses. The appraisal conditions shall be collected in superscript, the player is specified in subscript. The following is ideally expressed using simultaneous sub and superscripts. Let

|| T || X,PRE,END  =  t  X,PRE,END

be X’s pre-game appraisal of his end-game overall payoff. The game might proceed as follows:

  1. X believes he has a means of killing Y and getting away with it. He contemplates a Temptation payoff tX,PRE,END.
  2. At mid-game the Protagonist has made his move (he has committed the murder) but the Opponent has not yet reacted. X might be getting nervous. He is to receive either a Temptation or a Punishment payoff. Actually his plan goes awry and he is found out.
  3. At the end-game the Opponent Ω has acted and X is executed as punishment. Then, if X had produced no offspring and Z were measuring by the number of progeny, he could write
    p Z,END,END  = 0
    . In words: Z’s end-game appraisal of X’s end-game payoff is zero.

Of course, with a cost set including time, no payoff vector can be zero. Any transaction takes time. This notation should be equally applicable to commonplace games like the Buick Sale, Dollar Auction and Volunteer’s Dilemma (Poundstone, 1992).


Axelrod, R. & Hamilton, W. D. (1981). The evolution of cooperation. Science 211: 1390-1396.

Moir, A. & Jessel, D. (1989). Brainsex: The Real Difference Between Men and Women. London, UK: Mandarin.

Parker, G. A. (1978). Selfish genes, evolutionary games, and the adaptiveness of behaviour. Nature 274: 849-855.

Poundstone, W. (1992). The Prisoner’s Dilemma. New York, NY: Doubleday.

Sheppard, S. G. (2002). The Tyranny of Ambiguity: An Account of the Development of a System of Behaviour Analysis Called Procedural Analysis. Hull, UK: The Heretical Press.

Smith, J. M. (1982). Evolution and the Theory of Games. Cambridge, UK: Cambridge University Press.

Trivers, R. L. (1971). The evolution of reciprocal altruism. Quarterly Review of Biology 46: 35-57.

Wolpert, L. (1992). The Unnatural Nature of Science. London, UK: Faber and Faber.

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