Game theory: more players, more fairness?


Hi all,

I’m writing about the ways a multiparty system would work better than the two-party duopoly we’ve got in the US. I am looking for some game-theory expertise, and thought maybe someone here could point me in the right direction.

It seems to me that when you have multiple “players” (political parties) in, say, a congressional setting where collective decisions need to be made, there are stronger incentives towards setting rules that are fair. Whereas with only two “players”, there is more possibility of collusion between them to exclude certain kinds of maneuvers, etc., and/or there may be a cheapened version of fairness such as agreeing to take turns hijacking the system.

(This hijacking is also facilitated by the fact that with only two parties, one will have a majority unto itself in every legislative chamber. Whereas with more than two parties, not only would it be rarer for one party to have a majority on its own, but norms against hijacking would be strengthened, over time.)

So let’s call this the “more players, more fairness” hypothesis. I’m looking for a theoretical proof.

Anyone know of something like that?



Hi Dan! Welcome to the CES Forum.

This may be a good question for @Jameson-Quinn.

I think the question you’re asking depends on quite a lot of implicit assumptions on how congress operates already. In a game theory setup you can initiate a number of players, but then you also need to establish the rules for making rules. In congress right now that’s managed by the Rules Committee.
There becomes a recursive effect.

Also, in any game theoretical setup you need incentives for players. What do players want out of a system?
The question you’re asking is very meta. When players can edit rules you have a lot of recursion, which can destabilize the whole system and lead to chaos, otherwise known as game over.

Even in multi-party systems blocs emerge, which function to lower the number of players to the “pro” and “con” sides of a proposed rule. This is why, without analysis, I’d disagree with your hypothetical, though this is nothing along the lines of a proof. I’m looking forward to the further discussion.


Hi Felix, thanks for your thoughts!

I agree that the question is meta, and that blocs form in multiparty systems. But I think for rule-making purposes (as opposed to policy votes), that wouldn’t make too much difference. I think party, not bloc, is the right level of analogy (generally, though maybe with some particular exceptions) to “player” for purposes of setting rules – because coalitions can change over time, and each party has an institutional interest of its own.

As I understand it, at the beginning of each House (every two years), the Representatives vote on their rules. Usually there’s not much change, they just readopt most of what existed in the last House, and the majority party manages the process on its own because a majority vote is all that’s needed. (In the Senate there’s some controversy about whether a majority vote would suffice.) But given a multiparty system, this process could be very different. There would be an opportunity to drastically reform rules, right down to the structure of the committee system itself.

One might say the “recursion” you mention “bottoms out” with what’s in the Constitution, in Clause 2 of Article I, Section 5: “each House may determine the Rules of its Proceedings” – implicitly, every two years, by majority vote. That seems to make it pretty wide open.

As for incentives, I think the players want their policy preferences implemented, and they want electoral success. And so, I would think they’d also want procedural fairness, especially when there are 3 or more of them – that’s the hypothesis.

I have a hunch John Rawls’ “justice as fairness” may come in handy here, and I’m going to be looking into that.


FYI & FWIW – it’s not game theory, but I’ve found empirical support for the hypothesis in the literature of comparative political science:

How Parties Create Electoral Democracy, Chapter 2
Author(s): Royce Carroll, Gary W. Cox and Mónica Pachón
Source: Legislative Studies Quarterly, Vol. 31, No. 2 (May, 2006), pp. 153-174

Excerpt: “Second, the more concentrated the seat shares within an assembly
become (the smaller the effective number of parties becomes), the larger
the majoritarian bonus becomes (if we hold constant the age of the
democracy). The estimated effect is largest for cabinet portfolios but
still significant for committee chairs and board seats. This pattern is
our main result and is consistent with the evolutionary perspective we
have laid out, in which larger parties push for larger majoritarian
bonuses and smaller parties push for smaller ones.”

(“Majoritarian bonus” is a measure of disproportionality in the allocation of government responsibilities. And proportionality is a good proxy for procedural fairness, I’d say.)