Despite quite a variety of demands for quick and decisive financial aid for Greece, the EU countries and particularly Germany appear to act rather slowly at best. This non-reaction already made some commentators call for the IMF to step in. In this post I will try to analyze the situation using some basic game theory in order to explain how the observed actions can be seen as perfectly rational decisions instead of guideless waiting and to look for possible solutions. Of course, the following analysis will rest on massive simplifications and lack of formalization but I think, it can still add some valuable insights into the current situation.
Let the others act – the sequential game
To start the analysis, I would like to introduce a very basic sequential game in order to illustrate how inaction can arise even though all players would prefer seeing a bailout of Greece. In this game we have two players, for example Germany and France, which are considering whether bail Greece out or not. Both players would like to see a bailout of Greece ideally without bearing the costs. However, if faced with the decision either to help Greece or see no help at all, they would still accept the costs.
Provided these assumptions the game tree can be drawn. I added numbers from 1-5 to make the order of preferences clear (of course you can substitute them by more general algebraic terms but that wouldn’t really add much information).
The small arrows designate the equilibrium choices. The equilibrium path of the game would see Germany not acting and France acting. Now, what does this tell us about the current situation? The answer is that it shows the essential problem: no country wants to move first and make a commit before it doesn’t know the others’ decision. Just add more players to the game tree and you will see that except for the last player nobody will bailout Greece.
Attaining Critical Mass – the collective action game
The sequential game already gave us a first idea why not-acting might be a rational strategy for the European countries: its about saving the costs of doing it alone. But let’s have a little more sophisticated look at the situation. In this part I assume that all players act simultaneously or, what might sound more realistic, without knowing what the others are doing. Such a situation is called a collective action game.
I still assume that every player gets a certain benefit from helping Greece but will also face some costs. In addition to the situation above, however, I assume that the benefit grows the more countries are participating while the costs decrease at the same time. Now, the big problem of collective action games is bandwagoning or shirking. If a country does not participate but the others do, it will still gain the benefit but not pay the costs. In that sense this situation is still similar to the basic game above.
If one formalizes these assumptions, the following equations can be derived: whereby p(n) denotes the payoff for participating and s(n) the payoff for shirking:
Thereby p(n) denotes the payoff for participating and s(n) the payoff for shirking. The letter “n” stands for the number of players. For finding the Nash Equilibrium of the game, the following equation needs to be solved:
Solving for n yields
This number is greater zero, iff the costs are higher than the benefits. This seems reasonable to accept. In fact this n is not the Nash Equilibrium but the intersection of the two payoff function. However, knowing the intersection point is crucial for solving the game. I will not go into more formal details here but just do it graphically. The graph below shows the situation.
One can see both the payoff function for helping and the function for shirking. The equilibria are located at both extremes: either all help or noone. Only at those two points nobody has an incentive to deviate. On all points to the right of the intersection, participating yields a higher payoff than shirking thus creating an incentive for every country to become a participant (one can formalize this dynamic using evoluationary game theory but I will spare you the details).
What is important is that a certain critical number of helping countries is needed in order to get to the all-help equilibrium. In a situation with no country helping Greece, there is no incenctive to be the first to help. To put it in a nutshell: Europe needs leadership! Some countries have to make the first moves in order to make all better off (note that the all-help equilibrium has a higher payoff than the other).
Let me sum up the argument. The first game showed the rationale behind the currently observable inaction of the European countries in the question of Greece. The second game then further underscored the problem that basically nobody wants to be the first to help Greece. But the second game also showed how to overcome this stalemate. The answer is leadership by some countries in order to sufficiently lower the costs (one might also interpret it as the risks) of engagement for the other countries. This might then trigger a movement towards a all-help equilibrium.
