I am currently attending the very enjoyable Modelling Biological Evolution 2015 at the University of Leicester, where I have, of course, enjoyed catching up with old friends, meeting some great new people, and hearing some inspirational talks. This has led me to revisit an idea for a model for evolution of cooperation that I came up with last year. If I am being honest, I am unlikely to have the time to analyze these ideas. So I am putting them in this blog post in the hope that someone somewhere might be interested in doing the maths/simulations. If you are interested, please be in touch! It starts with a true story from.

Last year I was in a park with my two young children (aged 3 and 1) and some friends of the 3-year-old. Our 3-year-old and another child (also 3) started to argue about who was going to push the baby swing containing the 1-year-old. I discussed the situation with the children and the other child came up with the following solution: one child pushes the swing; the child who is not pushing the swing can at any point say “swap” and then they swap; repeat until they’ve had enough or we leave the park. At first I was dubious about what whether this would work – but what happened blew me away! The other child had the first time. My daughter swiftly said “swap” so she stared her turn only shortly after the other child called “swap” again. I said to my daugher: “maybe if you let X have a long turn, he might then give you a long turn too” and this is exactly what happened.

As we left the park I realized that this might be a very interesting game theory model! It has the possibility for cooperation, defection, reward, punishment, and then I realized a lot more (as I will explain below). It is also interesting for operation in continuous time rather than discrete episodes. I am not aware of other continuous time models although I would expect that they must exist. So here is my thinking:

1. The game is played for some time T (more on that later). The children take turns to push the swing and have a pay-off that is some function of the time spent pushing the swing (again, more below). The bystander can call swap – so the “stategy” lies with the bystander that will be some function to determine when they call “swap”.

2. It is immediately apparent that a linear pay-off for swing push time t won’t work (each child sums the time spent pushing the swing) as this could lead to a strategy of calling “swap” after an infinitessimaly small time interval. This might be fixed in one of a number of ways – two spring to mind: (i) Having a penalty associated with a “swap” since there is time lost to the game (so total possible pay-off to both players is reduced) (ii) Having a non-linear pay-off for swing time t, for example a sigmoid function (t^2/(1+t^2)) or such-like. This is probably more interesting. The key question is what constraints on pay-offs are needed so as to obtain interesting playing strategies.

3. The game could be constructed with either deterministic or stochastic strategies. Stochastic strategy is probably going to be more interesting.

4. What I would find particularly interesting is to model how the children’s knowledge of T affects the extent to which they cooperate. For example, the children might know T with absolute certainty, or T might be stochastic, with children knowing some partial information about T. Suppose, for example, T has some distribution with a certain mean and variance, and the children have this information. To what extent does knowledge of the variance impact on cooperation? This is interesting because it relates to reliability of parental information. Imagine a parent who says “you have 10 minutes left” and the child knows that this means 10 with variance 1, as opposed to a parent who says “you have 10 minutes left” and the child knows that the variance is very high (say 5): does this impact on cooperation? Similarly skew of distribution (right-skewed – so 10 minutes means at least 10 minutes vs symmetrical – so 10 minutes could mean shorter).

5. There is also the question of asymmetrical knowledge of T. This could be, for example, two children who have better or less capacity to estimate T (for example an older child playing with a younger child). Would asymmetrical knowledge lead to “cheating” strategies being admitted? Anyway, these are just some of my ideas. Over to you!

It’s a really cute idea, Dov. Can you think of any natural biological processes that have the same general strategic structure? That would provide a big incentive to develop the model fully. I don’t know enough biology to think of a biological process that might be modelled by your Swing game.

Of course, when it comes to using a swing, you don’t strictly need anyone to push you — you’re not dependent on a co-player. A while ago, for an introductory paragraph in a paper we were writing, I and my research group spent hours trying to think of familiar human activities that you absolutely can’t do on your own. I was surprised to discover how hard it is to come up with convincing examples, apart from sports and board games, of course, but they’re almost always strictly competitive (two-player, zero-sum) rather than cooperative. Even sex doesn’t really require a partner, and you could argue that dancing doesn’t either — it doesn’t necessarily take two to tango. Birds can’t remove parasites from the tops of their own heads with their bills, so they need cooperative co-players for allogrooming, but they’re not human. Eventually, for humans, we came up with teeter-tottering (see-sawing). Unlike using a swing, you can’t even begin to use a teeter or on your own.

Andrew Colman