# Majority with veto rule versus unanimity

In many group decisions unanimity is required to ensure that a reform will be adopted only if it benefits all its members. Multinational organizations may be the best example of this. However, when members have incomplete and private information about the convenience of the reform, a majority rule with veto power performs better in the sense that it would be preferred to the unanimity rule by all participants before they acquired their private information. This is the thesis in the work by Bouton et al. (2018) 1.

Consider the following example. There are three agents who have to vote on whether to adopt a given reform or to keep the status quo. The reform can be good or bad for all of them with equal probabilities. If they knew it is good, all would vote for it, otherwise they would vote against. All three would dislike a mistake in either direction. Before the vote, each agent can get some private information that basically indicates that the reform will be good or bad. However this information is not error-free. With probability two-thirds the information is correct. Also, the information of one agent is independent from the information of the others. Thus, if the reform is going to be a good one, the probability that the three agents receive a positive information is 2/3×2/3×2/3 = 8/27, the probability of two positive and one negative is 3×2/3×2/3×1/3 = 4/9 (notice that the first three in the multiplication is necessary as the negative report can come from any of the three agents), the probability of one good report and two negative is 3×2/3×1/3×1/3 = 2/9, and, finally, the probability of three negative reports is 1/3×1/3×1/3 = 1/27. Similar calculations show that if the reform is going to be a bad one, the probabilities of three, two, one and zero negative reports are 8/27, 4/9, 2/9 and 1/27, respectively. Using Bayes rule one can check that if there are two positive reports, the probability that the reform is good is 2/3: the probability of two positives under the condition that the reform is good is 4/9, and under the condition it is bad is 2/9, given a proportion of 4:2 –or 2:1- in favor of the hypothesis that the reform is good). Similarly, two negative reports imply that reform is good with probability 1/3.

Thus, in the example, if there are at least two positive reports, the reform should be approved according to the expectations of the three agents. The agent that receives a negative report wants the reform to be approved if the other two agents received a positive report. If they were to vote based only on their private information, the agent with the negative report would vote against. Unanimity rule would imply that the reform would not be approved with two votes, while the majority rule would imply it would be approved.

In the example, all agents preferred the reform provided it was a good one. If we complicate it to make room for the possibility that some agent prefers the status quo even if the reform is a good one, the majority rule would not provide an improvement for such an agent, but a majority rule with veto would. With this rule, the agent that prefers the status quo will veto the reform, whereas the agent that prefers the good reform but receives a negative report will vote against it (or abstain), but will not exercise her veto power. Thus the reform is passed if and only if it is satisfactory for all members.

The example illustrates the two results in the work by Bouton et al.:

(i) Majority rules with veto power are preferred to unanimous rules by all voters, and

(ii) Majority rules with veto power are ex-ante efficient (they select the option that the voters estimate is the correct one before they know their private information) in a broad class of situations.

In the example, the majority rule with veto was efficient also after players learnt all the information. In more complicated cases, agents may differ about the goodness of the reform after they have complete information over it, but if they agree it is a good one with the private information at the time they have to cast their votes, it will be ex-ante efficient that they approve it.

There are many more details to generalize the example. One of the most important ones is to make sure that the agents vote “right” in an equilibrium, meaning that the strategic vote does not prevent the mechanism from working as intended. The agent who prefers the status will always be willing to veto. Only the agents who like a good reform are potentially problematic. If the positive and negative information are both equally precise, agents will vote according to the information they get. If there is asymmetry in the information, the agents who receive the less precise information may be indifferent to vote one way or the other. As long as the negative information is not very precise, agents will not use their veto power, but as after a threshold is reached, they will start vetoing with some probability. The key is to see that in case of indifference (yes vs. no, or no vs. veto), the equilibrium probabilities of voting one way or the other are adequate for the optimality of the outcome.

In the information-aggregation equilibrium of the general game, agent behavior can be interpreted as a combination of what the agent would do under unanimity and majority rule (without veto power). Veto indeed allows agents to reproduce any strategy played under majority rule or unanimity. In particular, they use the veto power to protect their private interest (which they cannot do under majority rule), and they vote against the reform (without vetoing it) when they have a negative, but nonconclusive, signal about it (which they cannot do under unanimity).

After solving for the equilibria, the authors are able to study the efficiency of the majority with veto rule, showing that agents prefer it over unanimity for an adequate choice of the number of vetoes necessary to actually veto a reform, that for all problems there is always a veto rule that is efficient, and that the veto rule with a majority of vetoes being necessary to actually veto a reform is asymptotically efficient as the number of voters tends to infinity.

In the authors words, in addition to their strong theoretical properties, the simplicity of veto rules makes them particularly appealing for real-world applications. As discussed, there are voting bodies that use this voting system or slight variations thereof. Still, many voting bodies use unanimity or consensus, including international organizations such as the North Atlantic Treaty Organization, the Council of the European Union on most sensitive topics (excluding the Common Foreign and Security Policy), and the Southern Common Market (Mercosur). The results suggest that (i) they should consider using a veto rule instead, and (ii) such an institutional reform should not encounter much resistance.

## References

1. Bouton, L.; Llorente-Saguer, A., and Malherbe, F. 2018. Get Rid of Unanimity Rule: The Superiority of Majority Rules with Veto Power. Journal of Political Economy 126:1, 107-149.