Saturday, October 5, 2013

100. Of Hawks and Doves in the Logic of Animal Conflicts

Continuing with our discussion of evolutionarily stable strategies (ESS), we now consider a case study. Following Maynard and Price (1973), let us consider a species that possesses offensive weapons capable of inflicting serious injury. Two types of tactics of conflict between individuals of the species are conceivable: conventional (C) which are unlikely to cause serious injury; and dangerous (D) which can cause serious injury if employed for long periods. C tactics may, for example, be threat displays like puffed up size or looking extra tall, without any actual fighting.

In a game-theoretic approach to the problem of understanding such behaviour, the conflict is modelled as a sequence of moves and countermoves by two players. At each time step, the possible moves are C, D, and R, where R denotes retreat. Suppose player 1 employs tactic D in a move, for which there is a certain fixed probability that it would seriously injure player 2. When a player is seriously injured, he retreats, and the game ends, with the other player becoming the winner. The winner gains mates, dominance rights, etc., resulting in higher chances for the propagation of his genes into future generations.

If a player plays D in response to a move C by the opponent, we speak of a probe or a provocation. If a probe is made immediately after an opening move C of the contest, it is said to escalate the contest from level C to level D. If a player plays D in response to a probe, we speak of retaliation.

There are payoffs to each contestant at the end of a game, which are a measure of the likely reproductive success of the contestant. The payoffs depend on three considerations: (i) the advantage of winning; (ii) the disadvantage of getting seriously injured; and (iii) the disadvantage to the species as a whole of wasting time and energy in the contest.

Maynard Smith and Price (1973) modelled five possible strategies of combat:

1. Mouse. True to its name, the mouse (or the dove, if you want to call it that) never plays D. If the other player plays D, it retreats immediately, thus ending the contest. Otherwise, it goes on playing C for a preassigned number of moves. This is a ‘limited war’ strategy.

2. Hawk. Always plays D. Fights till the end. The end comes either when he is seriously injured, or when the opponent retreats. This is a ‘total war’ strategy.

3. Bully. Plays D if making the first move, or if the opponent plays C. Plays C in response to D. Retreats if the opponent plays D a second time.

4. Retaliator. Plays C if making the first move, or when the opponent plays C; but plays R if the contest has lasted a pre-assigned number of moves. Retaliates with a D (with a high probability) if the opponent has played D. A ‘limited war’ strategy.

5. Prober-Retaliator. A ‘limited war’ strategy. Plays C with high probability or D with low probability if making the first move, or if the opponent has played C. But plays R if the contest has lasted a certain fixed number of moves. Reverts to C after making a making a probing move if the opponent retaliates, but ‘takes advantage’ by continuing to play D if the opponent plays C. Plays D with high probability on receiving a probe.

Thus the Hawk strategy is a total-war strategy, and mouse, retaliator, and probe-retaliator are limited-war strategies. The main purpose of the simulation study was to see whether individual Darwinian selection (as opposed to the controversial notion of group selection) would favour total-war strategy or one of the limited-war strategies. They could demonstrate that individual selection does explain the observed behaviour, without any recourse to group selection. A group-selection model explaining absence of escalated contests would go something like this: Escalated contests would result in serious injury to many members of a population, and would militate against the survival of the species.

The probabilities used in the model were as follows:

0.10 for serious injury from a single D play;

0.05 that a Prober-Retaliator will probe on the opening move, or after the opponent has played C;

1.0 for the case that Retaliator or Prober-Retaliator will retaliate against a probe (if not injured) by the opponent.

The payoffs were assigned as follows: +60 for winning; -100 for receiving a serious injury; -2 for each D received that does not cause a serious injury, and only causes a ‘scratch’. In addition a payoff varying between 0 and +20 was awarded (to each contestant who was not seriously injured) for saving time and energy; the payoff was 0 for a contest of maximum duration, and +20 for a contest of minimal duration.

Symmetric contests were assumed. That is, the fighting prowess was modelled to be identical for the two players in a contest, as also the payoffs; they could differ only in the strategy adopted. An asymmetric contest, by contrast, would be one in which the fighting abilities of the players can differ, and/or the value of the resources gained or lost (i.e. the payoff) is different.

The five modelled strategies define 15 types of two-player games. Maynard and Price (1973) simulated 2000 games of each type.  The table below shows the average payoffs to each player in each type of play or contest. The number in a given row and column is the pay-off obtained by the row strategy when the opponent plays the column strategy. For example, the average payoff for the Mouse row and the Hawk column is 19.5; this is the payoff to the Mouse when playing against the Hawk. And the average payoff to the Hawk when playing against the Mouse is at the intersection between the Hawk row and the Mouse column, namely 80.0.

Player receiving the payoff


Which of the five strategies is an ESS for this population? Suppose it is Hawk. Such a population should consist almost entirely of Hawks. Therefore we have to look at the Hawk column in the table. We see that Mouse-Hawk and Bully-Hawk are evolutionarily more likely possibilities than Hawk-Hawk. Therefore natural selection will increasingly favour the occurrence of genes (alleles) with Mouse and Bully tendencies, at the cost of Hawk genes. Thus Hawk or ‘total war’ is not an ESS. If the species has mainly Hawks, it will wear itself out fighting.

The table shows that Mouse is also not an ESS. Such a population will consist almost entirely of Mouse, and the Mouse column tells us that Mouse-Mouse will fare poorly compared to other combinations in that column, meaning that natural selection will take the population away from a ‘mostly Mouse’ situation. A Mouse-only population will be easily dislodged from its habitat by an intruding species. Similarly, Bully is not an ESS.

It can be seen that Retaliator is an ESS (see the Retaliator column), even though Mouse-Retaliator contests do as well as Retaliator-Retaliator contests. In any case, as Maynard Smith and Price (1973) remark, a real population would contain young, senile, diseased and injured individuals who will adopt the Mouse strategy for nongenetic reasons.

Prober-Retaliator is almost an ESS. Thus, individual selection can explain why potentially dangerous offensive weapons are almost never used in contests within a species. An ESS does, however, require retaliation: The contestants must respond to an escalated attack by escalating in return. Such results have a bearing on how we humans can survive as a species.

The ESS idea has been extended substantially by the inclusion of asymmetric conflicts, involvement of multiple species, and the possibility of learning.