27 Results of the hawk-dove games

This section summarises class outcomes for Game 1 (same partner) and Game 2 (changing partners). Use it to compare with your own observations and results.

These results are from the 2022 class, who played both games.

27.1 Game 1: the same partner for 15 rounds

The average hawkishness in Game 1 (the proportion of rounds in which Hawk was played) was 0.62 (averaged over players), but players differed.

Is this more spread than we would expect by chance? The lecture showed that a mixed ESS can be reached in two ways: a mixture of pure Hawks and pure Doves, or every individual playing Hawk with the same probability. If every player had played Hawk with the same probability (0.62) in each of their 15 rounds, chance alone would give a standard deviation in hawkishness among players of about 0.13. The observed standard deviation is 0.22, so these players really did differ in strategy: some were consistently hawkish, others consistently dovish.

Did the level of hawkishness change as the game went on?

No: there is no detectable trend across the 15 rounds (p = 0.9). Nor was there a jump in Hawk play in the final round (round 15: 0.62), even though there was no next round in which to be punished.

27.1.1 Does being hawkish pay with a fixed partner?

With the payoffs used in this game (B = 4, C = 3), a single, anonymous encounter has a simple answer. Because B > C, Hawk beats Dove regardless of what your opponent plays (Hawk-vs-Dove gives 4 > 2 for Dove-vs-Dove, and Hawk-vs-Hawk gives 0.5 > 0 for Dove-vs-Hawk), so the one-shot ESS here is pure Hawk, not an intermediate mix. (The stable mix of Hawks and Doves, with Hawk frequency p* = B/C, only arises when the cost of fighting, C, exceeds the value of the resource, B: the opposite of the situation here. With B > C, the payoff ranking Hawk-vs-Dove > Dove-vs-Dove > Hawk-vs-Hawk > Dove-vs-Hawk has the same structure as the classic Prisoner’s Dilemma, with Hawk playing the role of “Defect”.)

This has an important consequence: within a pair, the more hawkish player always earns at least as much as their partner, because in every round Hawk earns more than Dove against the same opponent. So asking “who won in each pair?” always favours Hawk. The interesting question for Game 1 is a different one: do pairs that play Hawk a lot end up worse off, both of them, than pairs that hold back?

To answer it, we look at each pair as a unit: the pair’s mean hawkishness (both players’ cards) against the pair’s mean payoff per player per round.

The relationship is strongly negative (slope -1.96 per round): pairs that escalated a lot got stuck in Hawk-Hawk rounds worth only 0.5 each, while pairs that held back earned close to the 2 per round of mutual Dove. (The data file records players rather than pair IDs, so most pairs appear twice in this plot, once from each partner’s records, and the two points usually lie on top of each other. Because of this duplication we do not quote a p-value.)

That is the pattern expected if hawkishness provokes hawkishness: with a fixed partner, playing Hawk does not make you lose to your partner, but it lowers what both of you earn.

Reciprocity in repeated games

This idea is not covered in the lecture, but it explains why repeated play can change the outcome.

When you meet the same partner again and again, what you do now affects how they treat you later. This is sometimes called the shadow of the future. A simple strategy that does well in repeated Prisoner’s Dilemma games is tit-for-tat: cooperate (Dove) in the first round, then copy whatever your partner did in the previous round. Tit-for-tat is never the first to escalate, punishes escalation immediately, and forgives as soon as the partner returns to Dove (Axelrod & Hamilton 1981, Science 211: 1390-1396).

Two caveats. First, in a game of known, fixed length, defecting (Hawk) in the last round always pays, because there is no future to protect, so cooperation is not guaranteed in theory (in practice people often cooperate anyway). Second, with the class payoffs, taking turns (Hawk/Dove, then Dove/Hawk) pays exactly as well as mutual Dove, 2 per round each, so there is more than one way to cooperate.

27.2 Game 2: a new partner every round

The students were asked to compete against a new opponent every round. The idea was that this would make it much harder to learn your opponent’s strategy, and harder to come to an agreement that reduces aggression.

The average hawkishness in Game 2 was 0.69, a little higher than in Game 1 (0.62).

Again, hawkishness did not change systematically over the game (p = 0.72).

Finally, we can look at what the best strategy is in Game 2. Here each point is one player, and we use the mean payoff per round, because a few players missed some rounds.

The best strategy has changed: now the more hawkish players earned more (slope 1.21 per round, p = 0.006).

This matches the one-shot theoretical prediction much better than Game 1 did. In Game 2 you faced a new partner every round, so there was no opportunity to build reciprocity, and no way to be “punished” for playing Hawk against the same individual again. With no repeated interaction, no “shadow of the future”, the game reduces to the simple one-shot logic, and since B > C, Hawk pays off best.

In this setting we might expect natural selection to drive towards the evolution of more aggressive individuals.

27.3 Comparing the two games

Our hypothesis was that, compared with Game 2, players in Game 1 would play Hawk less often and earn more per round. In this class, hawkishness was slightly lower in Game 1 (0.62 vs 0.69 per player; p = 0.34), but mean payoff per round was not higher (1.25 vs 1.4; p = 0.36). Neither difference is statistically significant, so with one class of 18 players we cannot reject the null hypothesis. Remember too that Game 2 was always played after Game 1, so any difference between the games could also reflect learning or fatigue.

What did differ clearly between the games was which strategy paid: within Game 1, pairs that held back did best, while within Game 2, individual Hawks did best.

Put together, the two games illustrate a general result from game theory: repeated interaction with the same partner can favour restraint even in a game where Hawk wins any single encounter, while anonymous, one-off interactions favour the individually dominant strategy. This is one reason why reputation, memory, and repeated encounters matter so much for the evolution of cooperation, both in nature and in human societies.

Think about what this means in terms of human cooperation and conflict avoidance.

Can you see the value of understanding your opponent/competitor?