Hello /r/gamesandtheory,

Typically, this forum is a space for the discussion of high-level social engineering. With the mods’ permission, I would like to talk about something different: political engineering. Despite its name, this is not necessarily on the large scale of governments. After all, man is a political animal.

Although traditionally an academic subject, game theory in political science has uncovered exploitable flaws in even basic political systems. I present Condorcet’s Paradox, where the outcome of a basic decision-making/voting method can be manipulated/determined by whoever sets the agenda. This exploit is powerful because 1) the decision-making system feels intuitive, 2) the decision-making system can be applied to varied situations, and 3) the exploit works at all scales (and works best at large scales).

The basic and reasonable decision-making system is: if a democratic vote results in a tie, everyone lists their preferences and compares the results one by one, eliminating the losing option at each round.

Example: Three friends, Anne, Beth, and Charlie, are deciding between eating apple, banana, or chocolate pie. Anne likes 1) apple, 2) banana, and 3) chocolate. Beth likes 1) banana, 2) chocolate, and 3) apple. Charlie likes 1) chocolate, 2) banana, and 3) apple. Two out of three people prefer banana to chocolate, two out of three people prefer banana to apple, and two out of three people prefer chocolate to apple.

Let’s resolve this system of pie-based preferences through a simple majority vote. We first compare the options of apple and banana, and the winning option gets to go against chocolate. Two out of three people prefer banana to apple, so banana wins that round. At the second and final round, we compare banana to chocolate. Two out of three people prefer banana to chocolate, so banana wins round two as well. Option banana has won against all other options, so choosing banana pie is the democratic thing for the group to do.

The above example showed three friends choosing between three options. There are 216 potential combinations of preferences between the three friends, and a clear winner is evident in 204 of those cases, leaving 12 cases of gridlock, where no winner wins a sufficient majority.

Gridlock example: Anne prefers 1) apple, 2) banana, and 3) chocolate. Beth prefers 1) banana, 2) chocolate, and 3) apple. Charlie prefers 1) chocolate, 2) apple, and 3) banana. In the previous example, banana beat chocolate and apple, and chocolate beat apple. In that system, banana was the clear victor. Here, though, 2/3 of the people prefer apple over banana, 2/3 of the people prefer banana over chocolate, and 2/3 of the people prefer chocolate over apple. No option is more supported than any other.

In this gridlock, whoever controls the agenda of voting rounds determines the outcome. First examine the agenda where apple and banana face off in round one and the winning option goes against chocolate in round two. Here, apple wins over banana in round one since 2/3 of the people prefer apple over banana. In the second round, chocolate wins over apple since 2/3 of the people prefer it over chocolate. For this agenda, banana is chosen as the result of this democratic decision process. However, examine the agenda where apple and chocolate face off in round one and the winning option goes against banana. Here, chocolate wins the first round and banana wins the second round, making banana the democratic option. There is also an agenda that makes apple win, too. If Beth was in charge of agenda-setting, she should choose the first agenda as it would guarantee that her most favored pie, banana pie, would be chosen.

Thanks to rigorous game theory analysis, we know that any similar situation with preference gridlock is exploitable through agenda-setting. If you are in the above gridlocked situation and you can decide how to set the agenda of discussion, you can guarantee that the vote will go your way. These cases may seem artificial and limited, but as I said earlier, they scale up. With three participants and three options, gridlock only affects 5.6% of cases. But with hundreds of participants and options, the rate of gridlock rises to around 40% of cases. Even complicated situations can fall prey to this highly exploitable gridlock. Additionally, this kind of decision-making mechanism is flexible. The above examples were over the relatively silly issue of pie choices, but this kind of gridlock also affected Congressional decisions in the Revenue Acts of 1932 and 1938 (John C. Blydenburgh, “The Closed Rule and the Paradox of Voting,” Journal of Politics 33 (1971): 57-71) as well as the Tax Reform Act of 1986. Exploitable gridlock is the natural product of evaluating options with simple majority votes, and it is why congressmen and other major decision makers vie for the position of agenda-setter.

I am happy to discuss this and related topics in the comments. If there is sufficient interest, I may write another article on political engineering.

(Much of this was adapted from material presented in the lectures of Professor Kenneth A. Shepsle, professor at Harvard University and author of Analyzing Politics.)