Thomas T. Hills

1  Introduction

Any population of organisms that intends to improve itself (i.e., adapt) with respect to a given environment, may generally find itself at play in the fields of evolutionary dynamics. Wherein, any group of individuals enjoying the adaptive hat trick of heritability, variance among offspring, and differential survivorship will also very likely find itself-by whatever relevant performance measure it chooses-doing better in a given environment as the generations roll by. Such is the nature of adaptation.

The fact that numerous groups of organisms often succumb to extinction after eons of otherwise 'good' evolutionary work seems to be a bit discouraging. Contemporary dogma holds up both sides of the fence on this issue, that is, either something is going on and there is in effect some discernable force of species selection (Raup, 1991; Vermeij, 1996) or the vast majority of species persistence and abscission is nothing more than a crap shoot, all relegated to historical contingency (Raup, 1991; for similar tales about sex, see Buss, 1987). Either way, the act of adapting may in itself be risky behavior if the 'adaptive plan' (as presented by Holland, 1992) is unable to succesfully negotiate the peaks and valleys of the adaptive topography in both space and time. In other words, successful adaptation requires both an adaptive velocity, such that adaptations are achieved 'in time' to sustain the species (Lande and Shannon, 1994), and what can only be called 'adaptive foresight' in the sense that the phenomenon of canalization and Wimsatt's idea of generative entrenchment are simultaneously at work whenever adaptation takes a step on the landscape (Kauffman, 1993; Buss, 1987).

What follows is a rather juvenile look at a Brazilian dice game that exhibits prima facie the kind of risky evolutionary struggle that is akin to Maynard Smith's war of attrition (Maynard Smith, 1982). It may be useful to give the game a veil of biological relevance. Imagine if you will a forest of very short trees, all engaged in fragrant reproduction at the naive age of one. On this same beautiful day you are also aware of a rather aggressive northeasterly wind that, you easily imagine, could blow down any trees that gave themselves over to the myopic pleasures of height. But upon closer inspection you notice that here and there a few gambling flora are doing just that and have held out this first year of reproduction, and instead redirected their resources into growth. The higher they grow, they harder they fall, you imagine. But in the mean time, all the neighbors of these 'higher' trees suffer a shortage of light and have noticably small reproductive yields. You scratch yourself, pondering the fate of this child's forest.

In the forest above, the decision to reproduce at a given age is the strategy in question. Why shouldn't a tree wait a year? Waiting a year is almost always better than not waiting a year, especially if your neighbors choose to wait as well. But what about those winds? There is always some background risk that a given tree will be cleared of its present site, leaving nothing more than this years seeds to initiate its replacement. Neither waiting too long nor waiting not at all can be the answer to the forest's question. The answer, we will soon see, has questions of its own.

2  Ambition: the game

The Brazilian game of Ambition is played with a single six-sided die and more than one players. On a player's turn they may roll the die as many times as they like, granted that they don't roll a 1. However, if a 1 is rolled the player's score for this turn goes to zero and their turn is over. If they stop before rolling a 1, then they may add the sum of their rolls to scores from previous turns. A predetermined winning score is set before the game begins and the first player to reach that score wins. Apparently, another form of the game has it ending when all the players run out of money to buy beer or are otherwise too inebriated to read the die (we'll leave this particular version for the overly curious to investigate at home).

The game is remniscent of a number of biological challenges, in that the optimal strategy is affected by the strategies of others. One of our trees growing in the forest may choose a short stature at maturity to shield itself against structural damage during storms if their are no other competing trees. However, in the presence of other trees the success of a given height strategy will be affected by the strategies of neighboring trees. If reproduction is constrained by light availability, then growing to the canopy may be the only successful strategy, regardless of the potential risks from structural damage. Eventually, we imagine all successful trees will be playing the riskiest strategies, herded down the path of evolutionary vulnerability.

To investigate the phenomenology of Ambition, I will begin with a brief look at the easier insights, then investigate the coin flipping case, and then progressively move deeper and deeper into the jungle of escelating conflict, and dies of more appealing geometry.

3  Shaking Hands with the Basics

For the majority of this investigation I will assume that a player is playing against only one other player and that the other player's score is always unknown. Scores are compared at the end of one turn and the player with the highest score wins. Admittedly, the games of greater biological interest involve multiple players and have the nasty attraction of a stochastic time horizon-it always being uncertain exactly when the game will end. However, in the present context, this butchered version of Ambition will still grant us insight into the nature of escelation conflict.

A score-based rule will have the following format

= { p1 p2 p3 pk}
where pi is the probability that a player will roll again if they have a score equivalent to i. Notice that a player always rolls again with probability zero at p1, as a player who rolls one from the start has relegated themselves to an early doom.

Rules could also be formulated in terms of the number of times that the player has already rolled. For example, a rule of this form might be "roll five times and then stop." Rules of this form can be represented in a similar format as above, where pi represents the probability that a player will roll the die if they are rolling for their ith time this turn.

On may already realize that there are a number of ways to represent the rules that describe how a player should play the game. Game theory makes a clear distinction here between rules written in 'behavioral form' and rules written in 'strategic form'. For extensive form games, behavioral form strategies are the expression of mixed strategic approaches to games of pure strategies. In the [p\vec] case, we have written the rule in behavioral form, noting that a player must assess her current status during the play of the game. If she arrives at a score of three, then she will roll again with probability p3. Had she expressed her mixed strategic approach in what game theorists refer to as 'strategic form' then her only reference to probability would have come before her first roll of the die, when she would have probabilisticaly chosen out of a set of pure strategies.

The score-based rule [p\vec] is written in behavioral form. Pleasantly, for every mixed strategy written in strategic form there is a unique and equivalent strategy that can be expressed in behavioral form (Fudenberg and Tirole, 1991). Unpleasantly, the pure strategies in the game of ambition are represented by all combinations of zeros and ones in [p\vec] from p2 to p. That is, the a pure strategy may have a player rolling again with a score of three, but not with a score of two.

Rolling number strategies can be defined by expected outcomes based on probabilities of getting a number of consecutive non-zero rolls. For the six-sided die case the probabilities are

E(x) = ( 5
where x equals the number of times a player will roll. A player can expect on average a roll of four given that they do not roll a one. The probability of rolling a one goes up as a function of the number of rolls

Pr(x) = 1 - ( 5
. For rules based on the how many times a player has rolled in the past, the rules that optimize the expected average payoff will have the highest number of wins. By taking the derivative of the expectation function and setting it equal to zero, we find an optimum number of rolls at 5.5, with an expected average score of 8.07 per turn. A computer simulation of this game that competes alternative rules against one another verifies that rolling 5 or 6 times will always beat pure strategies that involve rolling a fixed number of times.

Figure 1: Imagine a nice picture here

For a given behavioral form score-based rule, [p\vec], the frequency distribution for possible score outcomes is

) =




i = 1 

This recursive formula follows from the likelihood that a player will flip heads consecutively until reaching a score of k and then stop with probability 1-pk.

To determine the frequency that one rule, [p\vec], beats another, [p\vec], I define a payoff function


) = k


where Qk([p\vec]) is the probability that a player ends with a score k or less playing rule [p\vec].

Using the above formulations as a model, a payoff matrix can be generated for two players playing Ambition with dice of any conceivable geometries.Towards the same end, a computer simulation was written in C that allows players to compete any roll or score-based strategy against any other roll or score-based strategy. These games were played out over 10,000 trials, and found to be show payoffs that accord with the payoff matrices generated by the above formulations. For zero-sum games of this type, we find that the payoff matrix is symmetrical. If a player can only choose to play or not to play at a given turn (probability that one rolls is either 0 or 1), then a roll again if score is less than n+1 strategy can always invade a roll again if score is n strategy. Paradoxically, an only roll once strategy can invade strategies that are too risky (roll again if score is less than ßome high number"). For dice with sides numbering greater than two, the payoff matrices all follow this same pattern. Trying to score a little more isalways a winning strategy, but trying to score too much more is fatal.

A winning roll number strategy has already been evinced. As expected, this strategy is inadequate against the best score-based strategies. There is a clear deficit of information in a strategy that uses nothing more than the number of times that one has previously rolled. A score-based strategy is more directly in tune with the performance measure at a given instance in time. Having rolled thrice, a player may have anywhere from six to 18 points. A score-based strategy would experience no such ambiguity.

4   Reduction of game

In the coin flipping scenario of Ambition, a player can choose to flip again if they flipped heads in all previous flips. However, a flip of tails reduces a player's score to zero for the turn and their turn is over. As long as no tails are flipped, a player may increment their score for the turn by 1 for each head that is flipped, and add this to their total score if they choose to stop flipping before flipping a tail.

This version is easily seen to be dominated by the player who flips but once,