Notes:

1. Some of the following mathematical expressions will be displayed approximately since simultaneous sub and superscripts cannot be obtained in HTML;
2. Omega, used to denote the whole environment, is unavailable in HTML and here U (the universal set) is used instead.

In the well-known work by Axelrod and Hamilton (1981) it is stated that "as one moves up the evolutionary ladder in neural complexity, game-playing behaviour becomes richer." It should not be surprising therefore if sexual behaviour, the 'battle of the sexes' being one of the oldest conflicts of all, is found to be a rich source of practical games. One might reasonably expect to find representations of Chicken, the Volunteer's Dilemma, the Largest Number Game, the Dollar Auction and Stag Hunt (Poundstone, 1992). Given the prevalence of parasitic mechanisms in nature it is implausible that such systems do not occur in man.

The Iterated Prisoner's Dilemma, which Axelrod (1984: 28) has dubbed "the E. coli of social psychology," is accepted as a basic model. It will be helpful to review the essentials of the game. It might also be appropriate to recall Jones' (1980: 7) assertion that "the subtleties of game theory lie more in the modelling than in the mathematics."

1. The Prisoner's Dilemma

With only a single interaction, defection is inevitable. This is because the biggest payoff is T, the temptation to defect if the other player is cooperating. Since T provides the highest payoff each player takes P, even while r > p. It is the inevitability of their respective defections which is the essence of the Dilemma.

This is illustrated by the scenario in which a protagonist exchanges a suitcase containing money for an opponent's suitcase of precious stones. There is to be only one round; the players will never transact again. One suitcase is placed in a field in Lincolnshire, the other in Yorkshire and they are to be collected simultaneously at an arranged time. Either could defect (say be filling their suitcase with bricks or paper) and in this game probably will. R is the profit of the transaction but T corresponds to obtaining the other player's suitcase only for the cost of its collection; P is the costs of a wasted journey plus a measure of guilt offset by the knowledge that the other player did likewise. Even though the moves are made simultaneously, the payoffs are asymmetric because each player will travel a different distance, make a different level of profit, feel a different amount of guilt, etc. Any alternating game (Nowak & Sigmund, 1994; see Table 2) is asymmetric if the opponent knows the protagonist's move; in this case there is inequality of information. The simultaneous game, like the symmetric game, is a special case.

Only when there is a prospect of future games is cooperation likely to occur. Then the accumulated reward payoffs can exceed a single, large temptation payoff. The probability of a future interaction is w. A condition of an iterated PD game is that r > ½ ( t + s ), so that play cannot proceed by the players politely taking turns to cooperate and defect.

The suitcase game may be iterated with cooperation by both parties and then it resembles the well-known one of Criminal/Fence. The authorities in this scenario probably qualify as an "adverse environment" or "common enemy" (Mesterton-Gibbons & Dugatkin, 1992).

Shortcomings of Existing Models. Procedures

The set of definitions proposed in Table 2 is not without its complications. According to Maynard Smith (1982: 10) and Axelrod (1984: 14), a strategy is a specification of what an individual will do in any situation in which it may find itself. Buss and Schmitt (1993) define strategies as "evolved solutions to adaptive problems." Here a procedure is an innate behavioural sequence by which an organism advances in its competition with a symbiont. A procedure should be capable of being modelled mathematically while a human strategy set, because of its complexity, is unlikely to be. Mesterton-Gibbons and Dugatkin (1992) have used Omega to denote a human strategy set.

A satisfactory definition of an evolutionarily stable strategy (ESS) may be yet to emerge, if only due to the difficulty in completely defining Omega. Smith defined an ESS as a strategy with the property that if all members of a population adopt it, no mutant strategy can invade. But "if stability requires a mixture of pure strategies, then individuals must adopt the appropriate mixed strategy; a genetically polymorphic population may be in an evolutionarily stable state, but, strictly, no individual is adopting an ESS" (Smith, 1982: 204). Dawkins (1989: 282) prefers to define an ESS as a strategy which does well against copies of itself. More recently Leimar (1997) considers that "it is important to apply the condition in a way that best represents the situation one intends to describe." Here it will suffice to define an evolutionarily stable procedure (ESP) as one which is advantageous (increases the net payoff; has a reason to exist) and heritable (evolutionarily viable; has a means to exist).

Because of its extraordinary robustness we can expect Tit For Tat (Table 3) to be an important feature of mathematical behaviour modelling. Variant strategies are continually being offered and a picture of the evolution of cooperative behaviour has emerged. For example, TFT can gain a foothold in an ALLD population but can then be displaced by more advanced strategies like GTFT and PAVLOV. However contemporary strategies have very limited memories; TFT, as others, is "memory one" (Nowak & Sigmund, 1993) so that, for example, the resumption of cooperation after a defection by one of the players takes no account of how many times the player has previously defected. In real life it is reasonable to suppose that policies will be chosen according to circumstance and that memory and discrimination will be important factors.

In attempting to model human behaviour formally we should be cautious about introducing variables and particularly of concentrating on entities which are interesting mathematically but which do not correspond to real-life phenomena, especially in the absence of a foundational model. One obvious, linear and indubitably important variable which is often missing from current models is time; an anticipated payoff can change solely with the passage of time (e.g. waiting for the repayment of a loan). However the perception of time might be altered to the advantage of a player (e.g. changing the clocks, maintaining that the only significant component is one's own lifetime).

Another important, basic model is kin-selected cooperation (Tit For Many Tats perhaps). Hamilton's (1964) rule has wider application in race-kinship, where relatedness may presently be difficult to quantify in game-theoretic terms but is clearly significant since human ancestors killed groups of one kind and interbred with others on the basis of unambiguous indicators such as hair colour and the ability to blush.

2. A Foundational Model: The Game of Murder

1. C/C' is the situation in which X cooperates by not killing, U cooperates by not punishing.
2. C/D' corresponds to the protagonist suffering the punishment for a murder unjustly.
3. D/C' occurs if X commits a murder but is not punished, for example if the authorities were non-existent or inactive, or if X were able to successfully disguise his defection (a common murder scenario).
4. D/D' occurs if X commits a murder and U delivers a punishment.

1. X believes he has a fool-proof means of killing Y and getting away with it. He contemplates a high temptation payoff t ^X_{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. In fact his plan goes wrong altogether and he is found out.
3. At the end-game the opponent U has acted and X is hung as a punishment. Then we might write p ^X_{END, END} = 0 (especially if X had begot no offspring). In words: X's end-game appraisal of the end-game payoff is zero.

Another Real-World Game

Interest is a two-player, asymmetric game in which a protagonist (the Borrower), who starts the game, wants to borrow some money from an opponent (the Lender).

A complete payoff decomposition is proposed in Table 5; it is claimed that every possible circumstance is contained within it. Disturbance of an existing sexual activity and Disturbance of an existing business activity may correspond to the 'standing' variable employed in some strategies (Table 3). 'Standing' is the reputation of a player.

1. C/C' is the circumstance of an interest-free loan in which the Borrower returns the money to the Lender as arranged, with no additional costs of procurement or collection. Then future cooperation is likely, and it is even possible that based on this initial game a larger loan might be advanced to the Borrower in the future, to finance some larger project. Or, when circumstances are reversed, their roles may be reversed also: the Borrower in this game may make a more sizable loan to the Lender in a subsequent one. (This would be a role-alternating game with increasing payoffs.)
   
   
2. C/CI' occurs with the circumstances set out above except that the Lender makes a charge (monetarily or otherwise) for the provision of the loan. The interest may be fixed, as in a commercial loan, or variable.
   
   
3. C/D' occurs if the Lender does not cooperate even though the Borrower wants to. He does not lend the money from the outset. The Lender's Temptation payoff LT1 might include the sensation of power and control he enjoys by being able to refuse (Enhancement of Self).
   
   
4. CI/C' can occur if a certain trick is played of having a drink bought whenever money changes hands, and this demonstrates how a lender can be manipulated into paying interest on lending money to someone rather than the other way around. In fact it is an example of manipulation by signalling. The Borrower returns a small interest-free loan in a café then, having won the Lender's trust by its return, and knowing that his opponent has money, he signals expectation of a drink being bought for him. Then if the Lender responds to the signal he incurs cost on lending the money.
   
   
5. CI/CI' models the situation of a loan being provided by a financial institution, with interest charged, and although the money is eventually recovered the effort required to collect it outweighs the benefit of lending it. In the language of Berne (1964) the Borrower is enjoying the benefits of playing 'Try And Collect.'
   
   
6. CI/D' could occur if incidental costs (such as for a meal) are incurred; with the Borrower playing CI the Lender is footing the bill. During the meal taken to discuss the prospective loan the Borrower divulges valuable commercial information to the Lender, who defects with the information.
   
   
7. D/C' occurs in the case of an interest-free loan which is not repaid.
   
   
8. D/CI' occurs in the case of a commercial loan which is not repaid.
   
   
9. D/D' occurs if the Borrower and Lender talk it through and agree in the end not to bother with the loan. The punishment for each party includes the time and energy expended and the information disclosed before the proposal was aborted.

The Table 6 model is convenient for illustration but C/C' is really the special case of zero interest, which is actually impossible; one cannot arrange a loan in zero time. In common parlance C/C' occurs when the interest is negligible. This game can otherwise be formulated as a two-by-two game with each player playing either CI or D.

3. Foundations of the Male-Female Game

The present work is a preliminary formalization of a new psychonomic system called Procedural Analysis. Some of the game-modelling concepts independently developed in this system have been anticipated by Trivers (1971), Parker (1978) and May (1987).

This analysis is not concerned with individuals particularly but with the distinct male and female evolution strategies. Rather than study animal models, which have been amply demonstrated, we direct our attention to human behaviour since human problems exist with greater force. One might cite the rise in male suicides, drug-taking and spree and serial killers as modern phenomena with adverse social effects. Consistently 97% of prison populations are male. Male unemployment is made more acute by an increasing proportion of jobs, especially sedentary and administrative posts, being taken by females.

Fundamental to the male-female game is that the costs of sex for females are enormous compared to those for males; many females would formerly die in childbirth or during attempts to induce abortions. Because of this phylogenetic balance of costs and benefits we might consider whether in some circumstances t^F > r^F but t^M < r^M. The female strategy is to raise the costs of sex while that of the male is to reduce them. Males compete, selecting for maximum fitness, while females conspire, instinctively acting together to raise the costs of sex and promote monogamy. The ultimate payoff for an evolution strategy is the number of progeny which results, including how successfully they in turn produce offspring.

The male is cast as opponent in this model because most sexual games are instigated by females, whether the male is aware of it or not.

Discussion. Female Strategy

As has been argued by Wolpert (1992: 128-135) and others, a good theory is one which simplifies, not complicates, and this has been the object of the author's recent investigations of human behaviour. Most things in nature are simple unless there is good reason for them not to be. An example of complexity due to necessity is the immune system, where a simple system would be inadequate because of the wide variety of concurrently evolving pathogens. However, simple processes are the most reliable because they are less prone to dysfunction. Dawkins (1989: 248-250) cites several examples of biological processes which are susceptible to elementary aberrations because the cost overhead required to prevent the error condition is so high compared to the probability of its occurrence. Examples are cuckoos laying two eggs in one nest (superparasitism), or a host bird's ability to recognize foreign eggs but not unrelated hatchlings, despite their inordinate size. Of course, parasitism is essentially a simple strategy, even though there is often considerable subtlety in its execution. The ant species Monomorium santschii usurps the hierarchy and manipulates the workers of an invaded colony for its own ends, and has implemented this stratagem so successfully that it has lost its own worker caste (see Shahak, 1997: 52-53).

This maxim – that most things are simple – is illustrated by the extraordinary robustness of the TFT strategy. A simpler strategy could hardly be envisaged yet TFT emerged the clear all-round victor in two computer tournaments conducted by Axelrod. Several of Axelrod's remarks about the TFT strategy point to it being females' default strategy, and in an iterated cooperative TFT game the payoffs for each of the players are similar. "The Prisoner's Dilemma tournament suggests that a good way for a player to appear untrainable is for the player to use the strategy of TIT FOR TAT... Using TIT FOR TAT is a good way of holding still and letting the other player do the adaptation" wrote Axelrod (1984: 153; his emphasis). TFT simply mimics any attempt by an opponent to improve on it, which is imitation. The observation was made that TFT "won, not by doing better than the other player, but by eliciting cooperation from the other player" (Axelrod, 1984: 137). This is an expression of manipulation: not doing something directly but getting someone else to do it. This can be regarded as the essential female strategy.

A behaviour was identified called Tit For Tat/Take Back which occurs when a protagonist retracts benifience from an opponent she considers has failed to reciprocate (e.g. retracting a gift). It should be possible to incorporate TFT/TB in the 'state space' or 'state machine' approach currently favoured by workers such as Leimar (1997) and Boerlijst et al. (1997).

While computer simulations employing various strategies have employed hundreds or even thousands of iterations, as has been shown, major games can be modelled as a single round. It is certainly true that in the initial interactions between individuals prospects can become very low very quickly, even after a number of game cycles (e.g. verbal exchanges) which is in single figures.

Summarized very crudely, because for one thing it neglects "subtle cheating" (Trivers, 1971), a general balance exists between cooperative behaviour and defection. Some procedures which might be employed to advantage in the long-term male-female symbiotic game are suggested and these are also means by which payoffs for one or both players might be modified. Specifically, it is proposed that a number of female procedures reduce male prospects. For example:

1. Increase the population, thereby reducing w, the probability of a further encounter.
   
   
2. Employ wide variations in dress and style, disrupting male targeting mechanisms and his ability to recognize the protagonist.
   
   
3. Erode the discrimination of the opponent, reducing his ability to foresee defection, identify the protagonist's policy or recognize defection. Perception does not have to be rational to be evolutionarily advantageous and humans appear to be capable of modifying their perception to be consistent with their emotions. The following forms of perception-manipulation exist:
   
   
   • a) Impose taboos and emotionalize language (once a subject cannot be openly discussed the perception of it can be altered to suit unstated goals);
     
     
   • b) Exploit humans' capacity for selective perception, such as the 'mind going blank' effect some individuals experience when faced with a page of mathematics, turning off one's attention when an embarrassing event occurs, or being unable to fully comprehend an event because it is inconsistent with established stereotypes;
     
     
   • c) Foster irrational perception (belief in astrology, UFO's, the supernatural etc.);
     
     
   • d) Promote distortions in magnitude perception or magnitude difference perception. Mesterton-Gibbons and Dugatkin (1996) discuss such distortions in perception in terms of RHP (resource holding potential) and quote Fechner's Law. Here it is proposed that:
     
     
     • i. Males make large differences larger and small differences smaller;
     • ii. Females make large differences smaller and small differences larger.
   
   
4. Reduce the opponent's prospects in subsequent encounters by attributing less significance to the material interactions (markers and handles) which have already taken place (see Appendix).

Beyond TFT

According to conventional opinion males are polygamous and females monogamous, but a highly advantageous female policy appears to be serial monogamy: the female becomes more adept at manipulation as her skills are applied each time in a different context. Such behaviour could be an example of parcelling (Connor, 1992); a further example of such parcelling might be limiting investment in a relationship by 'going Dutch.' Another basic premise of the male-female game is that the male is physically stronger than the female and, if provoked sufficiently, will use that strength. This truth, undoubtedly valid for the great bulk of human phylogeny, accounts for the evolution of a number of female procedures. The female optimally proceeds just below the threshold at which the male is provoked, and if she miscalculates things can go seriously awry.

In the adapted IPD/TFT model presented here the maximum long-term payoff for the protagonist would be obtained if she were able to successfully disguise that defection was taking place. If the opponent could be brought to a state of such confusion that he was unable to recognize defection when it occurred, the benefits for the protagonist in repeatedly taking the temptation payoff would be considerable. So advantageous are the cumulated temptation payoffs that we can expect Disguised Defection to be the dominant female policy.

The strategy of Disguised Defection would be facilitated if falsely elevated prospects were promoted or, in other words, an illusion was maintained. A practical expression of this procedure is a female maintaining a circle of hopeful suitors while she selects a single best match.

If Disguised Defection were employed in the suitcase game detailed earlier the protagonist might disguise his defection by filling his suitcase with counterfeit banknotes; the protagonist knows the money is forged but his opponent does not. Not only are the protagonist's costs diminished, his risk of arrest is much reduced by having the opponent distribute the forged notes. The forgeries are so good (or perhaps the notes are foreign currency, another common scenario) that the game continues for several cycles. It may further proceed to Overt Defection; perhaps the protagonist is having technical difficulties printing the banknotes and he may be tempted by another factor: he may not only wish to dupe his opponent but also want him to know he is being duped. This will of course disrupt the game but the protagonist may by this time have found a new, yet more susceptible opponent.

Explicitly stated, Disguised Defection models the female imitating the male, adopting roles and exploiting technologies which males have defined and developed once they become sufficiently facile, in order to disguise her defection from her natural functions. The perceived value of sex, and thus the costs which can be levied, are increased. It is conceivable that Disguised Defection has evolved from by-product mutualism in the absence of a 'boomerang factor': the probability that a non-cooperator is the victim of their own cheating (Mesterton-Gibbons & Dugatkin, 1992). An example of Overt Defection is miscegenation; most particularly, Occidental females bearing children for non-Occidental males.

Malign Encouragement can be employed by females to discourage males from using female policies against them (e.g. discouraging conspiring between males, encouraging competition). It is noteworthy however that there seems to be little to prevent males using female procedures against each other. Because of this and other restrictions on male policy (e.g. r^M > t^M) a realistic model of male-female interaction could resemble Table 8.

A number of devices can be employed by the female to discourage identification of her policy and this may explain why as basic a strategem as Malign Encouragement appears never to have been formally defined. Another important procedure is transduction: inducing a false feeling. Transduction typically takes place when a female quits a place because a certain person has arrived or an unpalatable fact has been stated or, more generally, when false guilt is induced. Creative Transduction is generating or inventing a problem for the purpose of blaming someone else. A rare survey of one expression of this procedure is published by Wilcox (1994).

Male Strategies

The analysis of human male strategies is a more daunting prospect. Since females are more uniform than males (Darwin, 1874: 340-350; Moir & Jessel, 1989: 89; Wilson, 1989: 107-108) they are easier to analyze. Females obfuscate but once this process is recognized the protection it affords becomes ineffective (at least as far as the analysis of their behaviour is concerned). An obvious alternate male policy is feigned cooperation but this is unlikely to be viable in the long-term.

Females possess superior acuity in the perception of signals, and if the male is obliged to be dishonest during sexual selection he may deceive himself in order to prevent detection and enhance his reproductive success. An obvious long-term female policy is to deny reproductive facilities to males who become immune to manipulation by females.

If the essential female strategy is TFT the male strategy, whatever it is, must be superior to it (otherwise we would now exist in an environment of ALLD). Crucial to the proper functioning of more complex strategies is discrimination and memory. Practical expressions of disrupting these components are discouraging discrimination on the basis of kind, thus discouraging players from "using more information about each other than is contained in the history of their own interaction" (Axelrod, 1984: 167), and rewriting history. Forgiveness is a mechanism by which prospects might be increased; unforgiving strategies, like TFT and GRIM (Table 3), revert to ALLD on a single defection by the opponent. A strategy analogous to "win-stay, lose-shift" (Nowak & Sigmund, 1993) may be consistent with male territorial instincts. Another important strategy which is played by males and females is Opposites: look at what the other player does and do the opposite.

References