Too Many Poor May Harm Redistribution

Photo: iStock/Hennadii

In a small two-class society democratic redistribution might only work if the size ratio between the rich and the poor is not too large, a study at the Institute shows. If the number of poor is larger than the square of the number of rich, in a game theoretical model the latter, although in a minority, push through their interests.

Imagine Entenbach, a village with 10 farmers and quite a few more harvest workers. Because the village has been able to build up reserves in recent years and is celebrating its 500th anniversary, the mayor announces at the celebrations that the community plans to waive the tax revenues from last year. Now the villagers are to decide: will everyone get their share of the tax back or will the entire pot be divided equally between farmers and harvesters? Both groups can cast their votes in the community hall punctually at 8 a.m. the morning after the festival. The majority decides.

Who has more at stake?

Since there are many more harvesters than farmers, it is very likely that they will determine the outcome of the election and that I can redistribute the tax cake, thinks the mayor, whose heart beats on the left. However, is he right? His prognosis only comes true if the number of harvest workers is lower than the square of the number of farmers, the economists Marco Serena of the Institute and Christos Mavridis of Middlesex University in London discovered with the help of a game theory model. Hence, if there are 10 farmers and less than 100 harvesters in the village, the chances are that the mayor will redistribute the tax revenue. If there are 100 plus harvesters, every-one will get back the share they paid. So the following does not apply: the larger the group, the more likely it is to win the election and assert its interests. This is partly due to the cost-benefit calculation that each voter makes for himself: The more harvesters, the smaller the share that the individual worker receives from a redistributed tax cake and the greater the temptation to have a lie in the morning after the feast instead of going to the polls. In contrast, the individual farmer has more at stake: Suppose a farmer has paid in two sheaves of wheat and two harvesters have paid one sheaf each. Full redistribution, where everyone gets a one third share of the four sheaves of wheat, harms the farmer more – he loses two thirds – than it benefits the harvester – he wins one third. Consequently, the farmer’s motivation to vote is higher than the harvester’s is, even though his probability of winning may intuitively appear lower to him. Apart from opportunity costs which take into account that voters have to give up something in order to vote, electoral research also knows the so-called information costs. These arise because voters have to find out which candidate or which party best represents them. In the setting studied by Serena and Mavridis, everyone knows how the game is played and what they have to win or lose.

Who is optimistic about their impact as a voter?

However, another consideration upsets this cost-benefit calculation. In the literature it is known as the paradox of voting. It says that for a rational, self-interested voter the expected benefits of voting are very likely to be less than the costs of voting. As the chance of being the one who makes a difference is minuscule, the individual can abstain from the election without consequences. As a matter of fact, this only applies to non-gigantic electorates, as the bigger the electorate, the smaller the probability of being pivotal, and the stronger become other motives for voting like ideological issues or the sense of civic responsibility.“Will my vote be pivotal and decide the outcome of the election?” the farmer and harvest workers ask themselves while turning over in bed at dawn. The answer to this question also depends on the size of the two groups and their size ratio. If they do not believe that their vote will decide the outcome, both the farmers and harvest workers would rather sleep than go to the polls. However, if they believe their vote might be pivotal they may want to vote, provided that the benefits exceed the costs of voting. If one takes into account all factors influencing voter participation in a game theory model and tries to determine the probabilities of all possible election outcomes, with an increasing number of voters the complexity also explodes. Game theory speaks of multiple equilibria; the election outcome cannot be predicted with certainty. Serena and Mavridis found a way to reduce complexity and calculate what had not been calculated before under the intuitive assumption that a small decrease in costs or benefits also incurs a small effect on the probabilities of voting. “There is a poverty trap lurking in the size ratio between the rich and the poor”, concludes Marco Serena. “If in a society with two classes the number of poor citizens is lower than the square of the number of the rich citizens, then the poor citizens may vote and redistribution has a chance of winning, otherwise the poor citizens abstain with certainty.”                                                                                                                                                                                                              chm

Mavridis, Ch. and Serena, M. Complete Information Pivotal-Voter Model with Asymmetric Group Size and Asymmetric Benefits. European Journal of Political Economy, forthcoming.

November 2020