As someone with an abiding interest in Science, you’ve probably often wondered what the best strategy is for stamping out dissent and bending the world to your will. Chasing after public cooperation is a bit like herding cats: you end up covered in tufts of fur and dander, with rivulets of blood coursing down your shredded arms.

What to do, Elizabeth Shue?

Attila Szolnoki (whose given name suggests that his parents were also interested in matters of strategy and enforcement) and fellow researcher Matjaž Perc (whose given name suggests nothing to me) have recently submitted a paper to the Journal of Theoretical Biology which may hold some answers. Szolnoki and Perc consider the evolutionary advantages of different punishment strategies in achieving public cooperation. They do this using a model called the “Spatial Public Goods Game.”

Fun for the whole family!
Might be fun, right? Let’s take a look at the basic gameplay.


Players take up positions on a spatial grid, and are assigned one of four strategies:
  • Unconditional Punishers contribute to the public good, and punish Defectors by levying a maximal fine. They bear the cost of the punishments themselves, which is proportional to the fines. 
  • Conditional Punishers contribute to the public good, and punish Defectors by fining them according to how many punishers (either Conditional or Unconditional) exist in the group. Like the Unconditional Punishers, they too bear the cost of the punishments themselves, in proportion to the fines. 
  • Defectors neither contribute to the public good nor to punishment. 
  • Cooperators contribute to the public good but do not punish Defectors.


Play proceeds as follows:
  1. Within each square of the grid, each of the contributing players (in other words, everyone except Defectors) contribute an identical amount to a common fund (the “public good"), which is then distributed equally to all players within that square (including the Defectors). 
  2. Punishers exact punishment (according to their two strategies) on all the Defectors within the square.
  3. Within each grid square, a player is randomly selected. 
  4. That player randomly selects one of their nearest neighbours and (a) gives them his/her public good share for that round; and (b) attempts to convert the other player to their strategy, with a probability of success that is super-complicated to explain, but in general means that richer players will have greater luck at conversions. 
  5. The process is repeated from Step 1 one million times, or until everyone is tired of it. 

Ha ha ha ha wow, that sounds like a hoot! So you may not want to drag this puppy out at your next party, but let’s see at least how Szolnoki and Perc’s games turned out.

Stupid, useless Cooperators
Well, in a sense it depended on the layout of the original grid. The figure below is from the submitted article. Dark green denotes concentrations of Unconditional Punishers, light green means Conditional Punishers, and red are those lazy Defectors. Cooperators aren’t shown, because “they are indecisive for the composition of the final state,” which I think means Cooperators are the boomerang-style Angry Birds of this game.

Figures (a) and (e) show two initial grid configurations. In (a), there are two, equally large, homogenous communities: one of Conditional Punishers and one of Unconditional Punishers. In (e), there is one heterogenous mixture of the two Punisher types. In both, the Punisher communities are surrounded by a sea of Defectors.  The three figures to the right of both Figures (a) and (e) show snapshots of progressive change to the strategy types on the grid over time.

Source: Szolnoki and Perc, 2013

In the case of (a), what you can see is that the Conditional Punishment community starts to fragment almost immediately: little fissures of Defectors start to show up everywhere. In contrast, the Unconditional Punisher community stays solid as a rock. However, it also starts to shrink like wool in a dryer. Before you know it—well, actually, by Figure (c), 6000 rounds later—those Unconditional Punishers have gone extinct. There are still Defectors in the Conditional Punisher community, but the Conditional Punishers have pretty much taken over the shop by the time we get to Figure (d).

Starting from the point of (e), we see a slightly different progression. First, the Punisher communities start to curdle, forming little islands of like-minded strategists, and by Figure (g) admitting large veins of Defectors. But by (h), while some Unconditional Punishers still exist, they are not significantly more numerous than they were in the initial Figure (e). Conversely, the Unconditional Punishers have again spread all over the damn place.

Szolnoki and Perc believe that their game shows that in both scenarios, the mild punishment doled out by the Conditional Punishers, together with the relatively lower costs they incur for doing so, makes them more resilient to the invading Defector hordes. Interestingly, the tendrils of Defector encroachment allowed by the Conditional Punishers, which initially seems like a weakness, “in fact forms the backbone of their deceptively simple yet very effective protection against further invasions.”

What have we learned? Well, that if you have no posse, it makes no sense to walk out like you’re Gary Cooper and it’s High Noon. That allowing opposing thought to exist within societies is not dangerous, but is rather a sign of strength. That the best way to take control of the game board is to bend and flex with circumstance, like a flexy bendy thing.

I think Atilla may be onto something there.