Tragedy of the Commons – Part 1

Edited 2020-10-27

‘Tragedy of the commons’, and it’s special two player case which is known as ‘Prisoner’s Dilemma’, is a game (or class of game) in game theory; which is interesting, both in it’s own right as a mathematical concept because it is so simple and yet has a surprising – even paradoxical feeling – result, but also because it is such a useful and natural model in evolution, economics and ethics, for understanding the limits of cooperation.¹

You may have come across the Prisoner’s Dilemma before, in some form or another. I’m going to skip the introductory motivational examples² for now and go straight to the abstract mathematical definition:
A game is a Prisoner’s Dilemma if it satisfies the following abstract conditions:
There are two players. Each player makes one decision – to either ‘cooperate’ or ‘defect’. Neither player knows the other’s decision when they make theirs³. The aim of the game as a player is to optimise the outcome relative to you⁴, where you rank the outcomes as follows:

    1. You defect and they cooperate
    2. You cooperate and they cooperate
    3. You defect and they defect
    4. You cooperate and they defect

Then it is a mathematical theorem, that in any game which is a Prisoner’s Dilemma (in the above sense), the optimal strategy is to defect. The proof is: Regardless of the decision made by the other player, the outcome where you defect ranks higher for you than the outcome where you cooperate.
What is surprising about this result, is that the outcome where both players play rationally and choose to defect is worse for both players than the outcome achieved by two irrational players who both choose to cooperate.

To make this more concrete, let’s consider the following example of a game which is an incarnation of the Prisoner’s Dilemma⁵.
In this game there are two players. Each player makes one decision – they can choose to either Take) Get £50 for themselves or Give) Get £100 for the other player. Each player writes down their choice, not knowing the other player’s choice. At the end the choices are revealed, and both requests are fulfilled (by some third party). Then, assuming that the aim of the game is to take home the most money, this game satisfies the conditions of a Prisoner’s Dilemma (where ‘cooperate’ corresponds to option Give, and ‘defect’ corresponds to option Take), since the outcomes are ranked relative to you as:

    1. You Take and they Give – you get £150 (they get £0)
    2. You Give and they Give – you get £100 (they get £100)
    3. You Take and they Take – you get £50 (they get £50)
    4. You Give and they Take – you get £0 (they get £150)

Hence by the same reasoning as above, or just applying the result we proved above in the abstract case, we know that the optimal strategy is to Take.

As a slightly less contrived example, consider the case of two neighbouring countries that each would like to take over the other country if the opportunity were to arise. Then if option ‘cooperate’ is to be unarmed and option ‘defect’ is to be armed, this situation satisfies the conditions of the Prisoner’s Dilemma since the outcomes are ranked as follows:

    1. You choose Armed and they choose Unarmed – your territory is increased, as you can invade the other defenceless country.
    2. You choose Unarmed and they choose Unarmed – your territory is unchanged, and no resources are wasted on arms.
    3. You choose Armed and they choose Armed – your territory is unchanged, but resources are wasted on arms.
    4. You choose Unarmed and they choose Armed – your territory is decreased, as you are invaded by the other country.

And so two rational countries should both choose to be armed, even though they would both be better off if they could agree to both remain unarmed.

More generally what the Prisoner’s Dilemma naturally models is any situation that requires “altruism” for cooperation to occur, where by altruism I mean behaviour that goes against immediate self interest. Such a situation usually takes the form of reciprocal altruism⁶: “You scratch my back, and I’ll scratch yours”. It relies on trusting that the ‘opponent’ will return the favour even though it is not in their immediate self interest to do so. It is important to note that this isn’t the case for all cooperation between two individuals. For example we can imagine a situation such as teamwork where there is a mutual benefit to cooperating, but it is not possible to exploit the cooperation. This can be modelled abstractly by the two player ‘stag hunt’ game, and for this game cooperation can be an evolutionarily stable strategy⁷. The ‘stag hunt’ may also be a more appropriate model than the ‘prisoner’s dilemma’⁸ for asymmetric exchanges such as trade, because it can be difficult to exploit the cooperation when the exchange happens simultaneously. There are also cooperation games like the ‘hawk-dove’ game, for which neither cooperation nor defection are evolutionarily stable strategies, and instead it is an equilibrium of the two strategies that is evolutionarily stable. The prisoner’s dilemma is unique, among games where cooperation leads to the greater good, because the only evolutionarily stable strategy is defection.

And yet the fact of the matter is that humans, and possibly other species, do consistently show reciprocal altruism. In particular, in a game show implementation of the money game example I imagine many people would choose option Give. So humans are able, in some cases, to ‘beat’ the Prisoner’s Dilemma and choose the option for the greater good over the selfish option. At first sight this appears to give justification to our intuition that human virtue must be a God-given quality that is an opposite force to self interest. But this isn’t the only explanation – it could be that humans are acting ‘rationally’ according to self interest, but they are not just trying to optimise the outcome of the game in isolation (this was an assumption we had to make in order for the Prisoner’s Dilemma model to apply in each situation) – they are considering it in the wider context (of their life) and in that wider context the game may no longer be evaluated as a Prisoner’s dilemma⁹.

Let’s consider the game show game as an example. In order to apply the Prisoner’s Dilemma model to this case we had to assume that the only thing the players were trying to do was optimise the amount of money they took home. This might be a close approximation to what they are trying to do, but it is not the only thing they care about; reputation is an important factor to consider, and in a televised game show can be enough to distort the outcome ordering and make the game no longer satisfy the conditions of a Prisoner’s Dilemma. To see the impact of the reputation factor, consider the following thought experiment where the relative impact of the reputation factor is decreased by either increasing the monetary prize or increasing the anonymity of the players (e.g. by making the game available to play anonymously online), and notice your temptation to ‘Take’ increasing…

The importance of reputation and trust as factors, suggests that a context in which cooperation could be ‘rational’ is repeat play – where you expect to play the prisoner’s dilemma multiple times with the same opponent¹⁰. And indeed this has been shown to be the case, by an experiment setup by Robert Axelrod in 1979. This experiment took the form of a computer tournament, in which participants submitted strategies, encoded as computer programs, that would play the ‘Iterated Prisoner’s Dilemma’ against each other. Out of all the strategies that were submitted, the most successful was a strategy called ‘tit-for-tat’, a very simple and intuitive strategy that starts by cooperating, and from then on does whatever the opponent did previously. The intuition behind the success of this strategy is that whenever two tit-for-tat players come across each other they can reap the benefits of cooperation, and yet unlike ‘always-cooperate’ players they can not be taken advantage of by ‘always-defect’ like players. It is unfortunately not possible to say anything general about the success of tit-for-tat against all other strategies, because in fact every deterministic strategy can be beaten by another in the right circumstances¹¹. But even just as an experimental result, it has value in demonstrating that ‘nice’ strategies can win in the end.

In the next part we’ll consider the natural extension of the Prisoner’s Dilemma to an arbitrary number of players – the Tragedy of the Commons – which can allow us to model group cooperation too. In a similar way we’ll see the limits of group cooperation and what conditions are necessary for group cooperation.