Thursday, 29 November 2012

Flooding and the weak link game

Britain is currently being battered by strong rain and heavy floods. I saw it at first hand on my morning walk around the valley: The stream in the valley was getting higher and higher with landowners and the council trying to do something to help the water escape. As I was watching these efforts, getting soaked by the rain, my thoughts turned to a hot cup of tea, and the weak link game.
    The weakest link public good game (also known as the minimum effort game) was first analyzed by Jack Hirshleifer in a paper published in 1983 in Public Choice. He began with the story of Anarchia – a circular island prone to flooding. To explain the story, imagine four people live on the island. Each person owns a portion of land and must decide how much effort to put into building a dyke on their land. If they all build a high dyke - the island cannot flood. If one of them builds a low dyke - the whole island floods. The name, weakest link, captures the fact that the minimum contribution is what matters. For example, see the diagram, if Carl builds a low dyke it doesn’t matter that Andy has built a high dyke – Andy’s land will flood.

Hirshleifer made a bold prediction –everyone will build a high dyke and Anarchia will be safe from flooding. To understand the logic, imagine that you are Andy, and suppose you know that Carl, Babs and Dawn will build a high dyke. You can build a high dyke and avoid being flooded. Or, you can build a low dyke, save some effort, but flood. It’s better to not be flooded, so you build a high dyke. In the parlance of game theory there is a Nash equilibrium where everyone builds a high dyke. Good news.
    Unfortunately, Hirshleifer was too optimistic. We now know from many experiments that people usually end up at a worse outcome. Again, put yourself in the shoes of Andy. This time, suppose you know that one of Carl, Babs or Dawn will build a low dyke. You can build a high dyke and flood, or leave a low dyke and flood. You might as well leave a low dyke and save some effort. So, there is a Nash equilibrium where everyone leaves a low dyke. Bad news. 
     And it’s the bad news that seems to win the day. The evidence suggests that people will not put enough effort into protecting themselves from floods because they expect that others will not put the effort in. Government intervention may, therefore, be necessary in order to build flood defences, even though it is in the interests of landowners to construct their own. 
     Let’s return now to my walk around the valley. The situation I observed was a slight twist on the weak link game. It was something like that pictured below. The stream runs down the valley and out into the English Channel. Instead of building dykes owners of the land can clear debris from the stream so that the water flows freely. If everyone clears the stream the water flows into the sea and no one floods. If one person does not clear the stream, then everyone upstream of the first blockage will flood.
What outcome can we expect this time? There is still the good Nash equilibrium where everyone clears the stream and no one floods. But, unfortunately, there is still the bad equilibrium too. To see why, suppose that Andy and Babs do not clear the stream. Andy’s land will flood, but there is nothing he could do about that because Babs did not clear the stream and so he was going to flood anyway. Babs, Carl and Dawn do not flood because the river is blocked on Andy’s land. No one has any incentive to clear the stream. The shred of good news is that only Andy gets flooded.  Babs, Carl and Dawn are saved by the blockage upstream!