Electoral College

This article is part of a series entitled Everyday Game Theory. See also:
1. The Escalator’s Dilemma; 2. Electoral College; 3. Passing Curiosity; 4. Lesson Time

This article is a response to the article Voting Cartels are Anticompetitive.

“How do you solve a problem like Ben Carson?”

Jim Rutenberg posed this question, in the March issue of the New York Times Magazine, before beginning an in-depth profile of the Republican presidential candidate and his role in the upcoming 2016 election. Though Carson, a retired Johns Hopkins neurosurgeon and an oft-called “outsider” [1] is not likely to be elected — as Rutenberg would have it — he certainly might disrupt the plans of the establishment Right. “A candidacy like Carson’s presents a new kind of problem to the establishment wing of the G.O.P.,” Rutenberg suggests. [2]

Ben Carson, a former pediatric neurosurgeon, is seeking the Republican 2016 nomination for president.

Ben Carson, a retired pediatric neurosurgeon, is seeking the Republican 2016 nomination for president.

The precise nature of this problem, though, depends on whom you ask about it. Ben Carson’s supporters might see things a different way. Negative claims about Carson’s electability – Rutenberg writes, for example, that “His chances of victory are miniscule” – could frighten Carson’s would-be supporters into the safer territories of the establishment. These threats could become self-fulfilling.

Ben Carson’s candidacy does exhibit a “problem”. The problem is unelectability. The solution is voter organization.

An example demonstrating why voter organization is not just permissible, but crucial to democracy

I’ll begin with a hypothetical electoral situation. (For ease of explication, I’ll use the names of actual Republican presidential candidates in my examples.) Suppose that an establishment candidate — Jeb Bush, say — has a healthy lead, with Ben Carson trailing in a distant second. A Carson supporter tentatively aligned with Jeb Bush would feel no qualms shifting his support to Carson.

But suppose, now, that Chris Christie — whom our voter dislikes — closely trails Jeb in the polls. The poll numbers might look something like this:

The hypothetical poll numbers in our thought experiment.

The hypothetical poll numbers in our thought experiment.

This Carson supporter might view the prospect of voting for Carson with significant trepidation. If he, and a few others like him, voted for Carson — instead of settling for Bush, a second choice — these voters might accidentally induce a Christie victory. (This might evoke the situation of Gore-Bush-Nader. That case, which differs from this one, will become important to us later on.)

In situations like this one — where, whether because of the media, polls, or a combination of factors, voters perceive one candidate to be unelectable — voters might shy away from a favorite candidate en masse.

This situation, worse still, can be both arbitrary and self-propagating. Say that arbitrary or irrelevant factors — such as name-recognition and political influence — catapult Jeb into the lead early on. It could happen that the bulk of those voters backing Jeb’s high poll numbers would actually prefer Ben Carson, yet remain “trapped” with Jeb for fear of yielding to Christie.

The tragic fact is that of all of our trapped Carson supporters acted together, they could successfully thrust Carson into first place.

The process of shifting would look like something like this. The shaded segments represent those voters undergoing the shift. I’ve included graphs representing the election states before, during, and after the shift.

As Carson supporters shift their vote, Chris Christie passes through the lead.

As Carson supporters shift their vote, Chris Christie passes through the lead.

This shift is unlikely to occur. Each voter, individually, faces the overwhelming threat of inducing a Christie victory, and is unlikely to act unilaterally. As a result, startlingly, our election could end with the victory of a candidate who, paradoxically, lacks the support of the majority of his own voters!

Here’s where voter organization comes in. Suppose that these would-be Carson voters could decide to form a collective compact, in which they mutually assure each other of a simultaneous change in vote. They would collectively enact the desired shift. An incomplete and partial shift would result in a Christie victory. But a full, cooperative shift would bring Ben Carson the Republican party’s 2016 presidential nomination.

This situation is a textbook example of what, in economic game theory, is called an assurance game. Voting for Jeb is risk-dominant, but voting for Carson is payoff-dominant. Here’s the catch: everyone has to take the risk together. If voters organize — and assure each other of their mutual cooperation — then they can confidently place their votes for the candidate they support.

This outcome, indeed, is exactly what democracy calls for. In an ideal democracy, each voter would vote for the candidate whom that voter prefers for intrinsic reasons. In practice, though, as we’ve seen, concerns about “electability” can yield traps in which voters are forced by risk aversion to vote for lesser candidates. By organizing, voters can overcome these obstacles, and place their votes where they’re deserved. They can restore democracy.

This article will give these ideas a game-theoretic formalization.

A preliminary remark on democracy and its problems

There are many bad ways to vote. People could vote for a candidate because they like his name, or his face, or because he supports a policy which benefits their special interests. Isn’t this part and parcel of democracy? When we accept democracy, don’t we also accept its consequences?

This criticism misses the point. It’s one question whether democracy itself is a good idea. Its dangers are obvious enough. Once we accept it, though, it’s reasonable to investigate voting behavior, particularly with an eye to identifying issues and proposing solutions to them.

This problem we’ll discuss below, in particular, is no simple result of accumulated individual folly. As we’ll soon see, it’s not just damaging, but also systematic, stubborn, and solvable. It’s systematic; predictable, logically consistent, and somewhat abstract, it emerges in a game-theoretic way from the collective interaction of rational agents. It’s stubborn; once it crops up it can be hard to escape, even for those who’d like to, and even for those who understand its innerworkings. It’s solvable; only through collective organization, though, can it be defeated.

These three factors make this problem, among those present in democracy, particularly worthy of our consideration. Attacking other problems — for example, seeking to educate voters who make irrational decisions — could demand indefinite resource commitments. This problem, because of its unique collective nature, might be solved through a small amount of judiciously applied social organization. This is an opportunity which should be taken.

Democracy sure has its problems. But that doesn’t mean we can’t make them better.

Thoughts on Voting Cartels are Anticompetitive

Josh, you’ve claimed that political organization is bad. I’ve claimed that it can be essential.

And I think we’re both right. I intend to show that the particular sort of political organization you’ve begun to describe — all those, in fact, featuring a certain set of subtle characteristics — are bad. Other types, meanwhile — such as those I’ve hinted at above — are good. The sort of political organization I’ll describe, finally, will turn out to be precisely the solution we need to the problem posed by the sort of political organization you’ve described. The situation will prove fascinating indeed. Let’s get started.

You’ve compared organized political movements to anticompetitive cartels among rival companies. As I understand it, your arguments can be roughly summarized as follows.

First, you discuss collusion among rival companies or union workers. You argue that, in market situations, individuals acting according to individual interest tend to produce desirable outcomes, and yet subcollections of economic actors may, by breaking away from the competitive pool and forming cartels, still further increase their own welfare while comparatively decreasing that of non-members. For example, companies (in the case of monopolies) or laborers (in the case of unions) can collude to simultaneously raise the price of the goods or services they offer. These cartels lead to social suboptima.

You argue that organized political movements function in similar ways. By voting in a block, Tea Party affiliates can increase the chances of a successful bid by an in-group candidate. Meanwhile, outside candidates are forced to accommodate the coordinated action of a large segment of the electorate. This organized behavior stifles competition and also produces social suboptima. The competition, here, is between political candidates.

The heart of your argument, of course, lies in this comparison. Though we’ll find issues, much of the comparison will remain intact.

The hard part will be what to do once we get there.

Is collusion really always bad? When can it be good?

The word collusion, strictly speaking, denotes simply working together, although it connotes something more sinister. By using this word, you’ve begun to answer your question before you’ve asked it. Working together — plain and simple — is not always bad. Is it bad here? We must examine the situation.

In the case of markets, the “state of nature” can be damaging and collusion can be helpful. To take the case of unions, we could imagine a case in which insufficient competition among prospective employers (say there’s only one) leads to suboptimally low wages. Workers could collude to raise their wages to optimal levels. Even among competing politicians, the “state of nature” could also be improved by collusion. We might imagine a situation in which voters demand a damaging public policy, and yet politicians collude to refuse to offer this policy. Standing strong to eschew it, one of these honest politicians will win. Natural competition doesn’t invariably lead to good, and collusion, rather than disrupting natural affairs, could actually ameliorate them. You should hold this in mind when using “cartel” as a pejorative.

Nonetheless, indeed, natural competition is often beneficial and collusion is often damaging, in both markets and politics. It’s natural to turn our attention to these sorts of cases. But why is political collusion, of the sort you’ve brought up, one of them?

We need to carefully explore the collusion you’ve described. It will prove to be quite complex.

Political “cartels” can indeed benefit members and harm others

We’ll start with a small thought experiment — one which has come up in our conversations.

Imagine a classroom with ten students. Every day, the class holds an election to determine the day’s recess leader. In sum, or on average, each student holds a 1/10 chance of winning. But suppose, now, that a band of six students forms a compact to vote only for each other. Each day, the group selects one of its members — perhaps on a rotating basis — and this member receives all six of the group’s votes. Because these six students constitute a majority, they may always determine the day’s recess leader. Now the group’s members enjoy the presidency 1/6th of the time.

This is a cartel, like those forbidden in business by antitrust law. The cartel subverts the class’s competitive voting mechanism for its own ends. Though each member faces a strong incentive to defect — after the lot has been drawn, even cartel members would prefer a free vote — by cooperating with the cartel, members ensure its continuation, as well as a larger shot at the presidency. Non-members, and overall group utility, suffer.

To make the comparison — between political and market collusion — very explicit, we can consider the example of bid-rigging among competing contractors. Imagine six construction contractors and a company accepting bids for a series of jobs. In the competitive situation, for each job, the contractors must underbid each other until they reach a market price. On the other hand, the contractors could initiate a compact whereby they pass around the privilege of an unobstructed bid. Every sixth of the time, some given contractor will bid as large as he pleases. The group of contractors wins and the company loses.

The classroom voting story will form the prototype for our real-life “voting cartel”. Things will become much more complicated in the general situation, and the analogy won’t always perfectly hold. But sometimes it will.

Here’s how the analogy might look if it held. A subset of competing republican presidential candidates aligns itself with a political movement — say, the Tea Party. Each Tea Party candidate has a corresponding constituency of supporters. In each election, through some mechanism — for example, by decree of the movement’s leadership, or even by a popular vote among members — one Tea Party candidate is chosen as the movement’s favorite. Once this happens, all other member candidates endorse this favorite, and these candidates’ constituent voters also shift accordingly. Though these voters would prefer a free vote, by participating in the cartel, they maintain the movement’s strength, and increase the likelihood that they, one day, will witness their own candidate on the movement’s banner.

If these voters vote for the Tea Party favorite simply because they prefer him to all other candidates, then the outcome coincides with that of democracy and there’s nothing to complain about. But if they vote for this chosen candidate — as opposed to, say, an outsider whom they actually prefer, or even an insider whom they actually prefer — not because of intrinsic democratic preferences but simply to maintain the movement’s strength and capture a majority down the line, then democracy is being subverted to a vote-sharing scheme.

In effect, members of such a movement instantiate a closed sub- political system within the larger political system. This sub-society plays by its own rules — these may or may not be democratic — and it controls (i.e., if it has a majority) or at least heavily influences the results of the larger election. This is a threat both to outside candidates and to democracy.

Richard, in particular, this goes towards answering the objections you raised at the end of this comment.

A few subtleties of political cartels, and how they differ from market cartels

The analogy between a market cartel and a political movement is not perfect. In the classroom example, the connection is somewhat direct. When we move to the real world, though, a few challenges will arise.

In the real world, “voting cartel” membership is beneficial only in certain special circumstances. In the classroom, each member of the pact — by joining — procures a clear improvement in his odds of victory, because every sixth day, all members now vote for him. (Key is the assumption that an individual supports his own candidacy.) Likewise, in the real-world analogue, a Tea Party voter — provided that he supports at least one of the movement’s candidates — stands to gain whenever his candidate is chosen by the movement. But things might not always work so well. Suppose now, for instance, that a classroom pact member no longer desires to be president, or, alternatively, that he does but that his classmates begin heedlessly voting elsewhere on his designated election day. This student will want out. Correspondingly, in the real world, a cartel member might simply find that he doesn’t much support any of a movement’s candidates. This voter might see little to gain by continuing to stand by a movement whose members he does not support.

To construct an extreme example, imagine a die-hard democratic voter evaluating the prospect of membership in the Tea Party’s “cartel”. The thought that this voter could stand to gain by pledging his vote to a member of the Tea Party’s lineup is ridiculous.

This is part of the reason why it’s strange to refer to these political movements as “voting cartels”. In a cartel, though defection is an ever-present temptation, cooperation still brings higher utility than nonmembership. (These are the prisoner’s dilemma’s so-called temptation, reward, and punishment payoffs, respectively.) In a “voting cartel”, though, it’s possible — as we’ve just seen — for membership to bring lower individual utility than nonmembership. In this case, not even the prospect of future [3] “cooperation” could induce an individual to remain loyal. It’s not that voters aren’t bound into their membership — they aren’t bound in market cartels, either — but rather that their membership might not even convey benefits.

The precise conditions under which membership in a voting cartel conveys benefits are somewhat complicated. Though it’s now necessary, as we’ve seen, that a member support one of the cartel’s candidates, even this condition is not sufficient. Supporters of a wildly popular candidate should decline cartel membership. Why should they agree to share their allegiance? Supporters of less popular candidates might stand to gain. If the classroom candidates belonging to popularity slots 2 through 7 (or even 5 through 10), for example, formed a cartel, they could increase their share of the presidency from 0 to 1/6th. Similar reasoning could apply to actual political candidates. I won’t explore too deeply here.

The power of a cartel also depends on its size. Again, the situation in politics differs from that of markets. In markets, cartels are most effective when they include all peers in a fixed market position. The most effective political cartel contains just over fifty percent of the voters. Smaller cartels aren’t guaranteed majorities; larger ones must share political influence among more candidates.

That these movements can exist, and that they can be damaging, is sufficient reason for our concern. In short, Josh, your article might be onto something.

Why risk-aversion might hinder the dissolution of dominant and unpopular cartels

We’ve seen that political cartels — like market cartels — can bring benefits to their members, and thus might arise naturally, and also that they can be damaging. We’ve also seen that, unlike market cartels, political cartels bring benefits to their members only under special circumstances.

This might be good news to us. Though market cartels can be economically harmful, they’re also unstable, and only through the emergence of cooperation in certain iterated prisoner’s dilemmas [3] can they endure. In political cartels, further still, collusion can cease to bring benefits, and we can imagine individuals defecting heavily from weak movements. We might believe, then, that voters tend to maintain support of a movement only when they support one of its candidates, and that, though this voting process isn’t exactly democracy, it’s not far off.

In reality, the problem is worse than this. Though political cartels can cease to bring reward, they might simultaneously continue to bring stability — and this stability can deter people from defecting even when they’d like to. This factor will make political cartels both harmful and persistent.

Let’s return to the example of the classroom. Let’s suppose that a member of the gang of six decides that he no longer wants to participate. He’s tired of being president — especially wrongfully — and he also recognizes the value of another genuine, bright student, a classmate who happens to be outside of the cartel. Let’s suppose also that a different student outside the cartel, determined and influential, aims to build an opposing cartel, and that our voter highly dislikes this second student. Our voter might be scared to vote for his friend because, dissolving his own cartel, he might hand power to his enemy. We might even suppose that a few other cartel members face similar thought processes. Though the cartel has ceased to bring these voters benefits, they find themselves bound by risk-aversion.

This situation has a parallel in the real world. In fact, we can recall the example of Bush-Christie-Carson. In this hypothetical situation, Bush represents the existing political cartel — the establishment — and Carson represents the desired outsider. Supporters of Carson aligned with Bush fear voting for Carson lest they hand power to Christie.

A voter’s decision to defect from a cartel, then, might be tempered by that voter’s feelings regarding who would gain power in the absence of the cartel. The identity of this next leader, of course, depends on the behavior of other voters in the cartel. We have a collective action problem on our hands.

Dissolving a dominant cartel, viewed as an assurance game

This is a classic assurance game. The formal definition is somewhat complicated [4], but, briefly, an assurance is a game with two Nash equilibria, one payoff-dominant and one risk-dominant.

Things are getting turbulent in the classroom. Imagine that our original cartel has shrunk to five members. A persistent and corrupt leader now forces all cartel members to repeatedly vote for him, and the remaining four would like to defect. The disliked classmate, meanwhile, has assembled an opposing campaign with four supporters, and the favorite friend stands alone and votes for himself. The unsatisfied members of the dominant cartel — who as above, let’s suppose, prefer the friend to the cartel and yet dislike the enemy — face the following decision process. If all four simultaneously shift their vote, they’ll propel the underdog into power. But if even one of the four fails to act, then the enemy will win (or tie) with four votes. Still worse, neglecting to shift carries the guaranteed benefit of preserving membership in the dominant (though perhaps now less-so cartel), while those who switch might find themselves out in the cold.

This description carries over with little change to the real-life situation of Bush, Christie, and Carson. In this case, the “players” of the game are those Carson supporters currently aligned with Bush.

Two of the above outcomes — all stay and all switch — are Nash equilibria, in the sense that in these situations, each player, given all of the other players’ choices, would decline to reverse his own. All switch, though, is better than all stay, in that players value it more. It’s also riskier. Indeed, that to maintain one’s allegiance with the dominant party is risk-optimal is intuitively plausible. Switching one’s vote, one risks dissolving the dominant party and his membership therein, and risk-aversion might keep one fixed. Making the definition of risk-dominance precise, unfortunately, gets complicated, and I refer the reader elsewhere [4].

The bad news is that in organic assurance games, people tend to get stuck in the risk-dominant equilibrium [4]. That’s where organization comes in. If players organize to assure each other of their cooperation, then they can influence their reciprocal reliability assessments in such a way as to attain the payoff-dominant equilibrium. In short, they can vote together.

This might sound strange. Aren’t we just creating another political cartel? Not quite. An opposing political cartel could certainly do the job here, though it wouldn’t be just, and, at any rate, it’s not what we’re arguing for. Political cartels are vote-sharing schemes in the above sense that members take turns voting for candidates whom they might not necessarily support. This proposal, on the other hand, concerns a political movement devoted to a single candidate. Voters enlist only if they intrinsically support the one and only candidate associated with the movement, and this is just democracy at work (see above). The collective aspect of this organization is necessary only for the voters to assure each other of their cooperation in an endeavor which they each individually support.

Bringing these thoughts to bear on modern elections

We can finally begin to bring our example home. In practice, information distribution mechanisms — including the media, advertisements, and publicly published poll numbers — can serve to instate widely held perceptions regarding the relative standings of an election’s candidates.

These perceptions are also influenced, of course, by the lingering perceived dominance of powerful political cartels. Political cartels, in fact, will remain important to us precisely to the extent that they influence these voter perceptions. In particular, perceptions to the effect that a certain candidate lacks popular support — which, recall, can induce all-but insurmountable obstacles to that candidate’s election — are effectively perpetuated by political cartels.

Cooperation can remedy these problems. Of course, the degree of explicit cooperation which would be required — even allowing the presence of sufficiently many would-be backers — to induce a material shift in the polls would be considerable. In practice, organizers would most likely operate by injecting their message into this “public information conduit”. By making their political movement known and respected, they might build subtle confidence among voters. And confidence differs from assurance only in degree.

These “political assurance movements”, again, differ from cartels in that they support a single candidate, and in that they earn the support of their members through the intrinsic quality of the candidate whom they represent.

Voting cartels are anticompetitive. And voting assurance movements can restore democracy.

A game-theoretic formalization

The conditions under which these assurance games have been said to occur might seem somewhat intricate and complex. In this section, I’ll attempt to construct a general “theory” of elections in this vein. Our assurance game will appear naturally as the crucial special case.

We can start by describing any election in the following abstract way. We focus on elections consisting only of a single “tier”, with no hierarchical run-off systems.

Say there are n total candidates. A particular state of the game consists of a common set of impressions held by every member of the public regarding the share of the popular support which falls to each of the n candidates. To be clear, this is a single, common, set of beliefs, and each voter shares it identically. We can think of these perceptions, and refer to them, as “poll numbers”, though, as mentioned above, other factors like media and advertisements also play a role. We can call this overall state an election setup.

Each voter, meanwhile, assigns a personal utility ranking to each of the n candidates. Unlike the poll numbers, these utility rankings depend on the voter.

Thus an election setup could look something like this:

A generic election setup with n candidates.

A generic election setup with n candidates.

Now the key idea is: When does a voter switch his allegiance? To analyze this, I’ll introduce two separate, related games, which we’ll explore “in parallel”. These games correspond to individual and group decisions, respectively. In both cases, for simplicity, we’ll suppose that the game operates in a sequential, or turn-based, way.

Both games begin with an election setup. In each move of the individual game, a single voter, who’s currently (i.e., according to the poll numbers) aligned with a certain candidate, decides whether to stay or to switch his vote to a different candidate. If an individual decides to switch, between candidates B and A, say, we say that he plays a B-A-switch. In the group game, we postulate, for each among all possible individual utility ranking schemes, an organizing body representing all voters with that particular scheme. We assume that groups have the power to coordinate and bind the voting behavior of their members. In the group game, the players are the groups, and in each turn, a single given group must decide whether to shift the votes of all of its constituent members to a single candidate. If it does so, to candidate A, say, we say that it plays an A-switch. After each move, the election setup changes accordingly, and then another player makes his move. The goal is to maximize utility.

The similarities, and, most of all, the differences, between these two games will reveal enlightening facts about politics and about the above discussion.

The Individual Game: A systematic exploration

Should an individual always choose to switch his vote to the candidate whom he prefers? Not necessarily. The key idea is that an individual — considering whether to take part in a proposed shift — must assess not just the shift’s beginning and ending states, but also its all of its intermediate states. Recall the situation of Bush-Christie-Carson. As voters aligned with the front-runner Bush begin to shift, Christie briefly obtains the lead, until at the end finally Carson recovers it. This framework — whereby we analyze, throughout all stages of the shift, who holds first place — will be instructive.

We’ll refer to a mass voter movement as a shift. In fact, we will also begin referring to shifts by the total sequence of candidates who, throughout the course of the shift, find themselves in the first place position. For example, we might refer to our above example as a Bush-Christie-Carson-shift.

At first glance, we might imagine that many candidates could pass through the first place position throughout the course of a shift between two particular candidates. The following result shows that this is not the case, and that, indeed, in a sense, the above example of three candidates exhausts all of the possibilities.

Claim 1: Reduction of shifts to the case of three candidates. Consider an election setup with n candidates, two candidates A and B, and a prospective shift of a subset of B’s voters from B to A. Let m be the ranking of the candidate initially-higher-ranked among the pair consisting of A and B, and denote by C the candidate in the (m+1)th position. (We allow that C = A or B.) Restricting our attention to the subset consisting of those candidates holding ranks m, …, n before the shift begins, then, throughout the course of the shift, the highest-ranked candidate among this subset of candidates will be at all times either A, B, or C.

This claim is most important in the special case when m = 1. In this case, one of the candidates A or B begins in first place, the restriction is vacuous, and the claim then states that throughout the shift, the first place position will be held at all times by either A, B, or the candidate immediately behind the leader among A and B. This explains why we can consider this claim as a reduction to the simpler case of three candidates. (In particular, we can again recall the example of Bush-Christie-Carson.)

Note that not all candidates A, B, and C need pass through the lead over the course of the shift.

I won’t bother with proving this claim, but its statement is plausible enough. If one of A or B is in first place, then as we shift from B to A, the leadership could stay within the pair, but if not — no matter what happens to these two (we could even imagine removing them entirely) — because all other candidates remain fixed, the prevailing leader can otherwise be none other than the candidate C trailing the leader among A and B. In our example, Christie plays this third role.

As we’ll see now, our statement can be made more precise. Because of our reduction, we now consider only elections with three candidates.

Claim 2: Characterization of the circumstances of B-C-A-shifts. Consider an election setup with three candidates, A, B, and C, and a prospective shift of a subset of B’s voters from B to A. Then, for the various holders of the first place position throughout the shift to be B, C, and A, respectively and in order, it is necessary that the curve consisting of the initial poll standings of candidates B, C, and A be decreasing and convex. If, in addition, the shifting subset of B’s voters is so large that A concludes the shift with the lead, then these conditions are also sufficient.

First, a few remarks. I’ve declined to exclude trivial cases involving empty subsets, non-strict leadership, and non-strictly decreasing curves. Also recall that in the Bush-Christie-Carson case, the curve marking these candidates’ initial standings was indeed convex.

The shift we described there was a Bush-Christie-Carson-shift, or, with proper naming, a B-C-A-shift. The claim says, in effect, that only in convex cases like this one is it possible to observe this passage of the first-place position through an intermediate third party.

Of course, convexity is a necessary, and not a sufficient, condition. Indeed, consider the example of Gore-Bush-Nader, as follows:

Though Gore-Bush-Nader is decreasing and convex, the Nader contingent is not large enough to propel Nader into the lead.

Though Gore-Bush-Nader is decreasing and convex, the Nader contingent is not large enough to propel Nader into the lead.

In this case, though the curve is decreasing and convex, even a united shift of all the shaded Nader supporters will fail to procure A’s — in this case, Nader’s — final lead. This is a Gore-Bush-shift; to use previous terminology, it’s a B-C shift as opposed to a B-C-A shift. (Spoiler alert: B-C shifts are a bad idea.) We could also, in fact, imagine a simple B-shift, in which the subset of B’s voters is so small that B’s initial lead is never broken. Thus what can go wrong is either for the shift to never start, or to fail to finish.

When none of these “small-group pathologies” occur, convex decreasing curves produce B-C-A-shifts, as the final part of the claim points out.

To “prove” this claim, picture, in the bar graph representing these three candidates, the shape of the bars as they undergo a transformation whereby members of the highest bar steadily shift to the lowest bar. In and only in the convex case, the initial difference between B and C is smaller than that between C and A. Taking a subset of the switching voters whose size resides between these two numbers (provided that the contingent of switchers is so large that this is possible), and moving this subset from B to A, we achieve the desired intermediate stage in which C places first. If, additionally, the contingent’s size exceeds even the larger difference between C and A, then A will end with the vote.

If we reverse the convexity assumption, then we can instead encounter B-A-shifts. Consider the modified example of Bush-Walker-Carson, in which Scott Walker, though between Bush and Carson, is closer to the latter:

ecause Bush-Walker-Carson is not convex, Walker, the intermediate candidate, never passes through the lead.

Because Bush-Walker-Carson is not convex, Walker, the intermediate candidate, never passes through the lead.

Though the curve consisting of the poll numbers of the three candidates is decreasing, it’s not convex. As the chart shows, during the Bush-Carson shift, Walker never takes the lead. This is a B-A-shift. Of course, another way to come across B-A-shifts is to remove the third candidate altogether.

We’re now in a position to offer a fairly complete description of the optimal play strategy in the individual game. Though all of the above results are entirely independent of any individual’s utility assignment scheme, it will now be necessary to refer to a voter’s preferences. Thus, now, we might name candidates according to their standing in some particular individual’s utility assignment scheme, and refer to such things as 2-3-1-shifts. Recall that in a descending sequence pq of ranked candidates, the candidate q with the smaller rank is better.

Claim 3: Individual play strategy. Consider an election setup with n candidates, 1, 2, …, n, and a particular voter V. Without loss of generality, assume that the candidates 1, 2, …, n are named in such a way that candidate i holds the ith position in voter V’s utility assignment scheme for all i = 1, …, n. Suppose that V is currently aligned with the candidate j, where j is some number 1, …, n. If, for some numbers p > q, with p possibly equal to j, V judges that the contingent of qsupporters in j’s block is sufficient to induce a p-q-shift, then V should play a j-q-switch. If, for some numbers p > q > r, with p possibly (and actually necessarily) equal to j, V judges that the contingent of r-supporters in j’s block is sufficient to induce a p-q-r-shift, then V should play a j-r-switch. If there are multiple such opportunities, V should select the (unique) one for which the final number is minimal. If there are none, V should play stay.

This claim says, in effect, that V should switch his vote to a better candidate when and only when poll numbers are such that if enough people did so, they would propel a sequence of successively superior candidates into power.

The fact of this strategy’s optimality rests on our previous results. Claim 1 permits us to substantially restrict the set of all possible salient cases which confront a player. The only possible shifts are p-shifts, p-q-shifts, and p-q-r-shifts, where p, q, and r are three distinct numbers. V should surely act when these numbers are increasing. On the other hand, p-q-shifts, with pq, and p-q-r-shifts, with p < r, should clearly be avoided. This leaves only the immortal p-q-r-shift, with q > p > r (e.g., 2-3-1). Because of risk-aversion, such a switch is inadvisable (see above).

I’ll offer a mild technical refinement regarding this last claim; this can be skipped on a first reading. One could argue that even switches geared towards inducing shifts of this latter type — p-q-r, with q > p > r — are individually viable, in the following way. Voters who are aligned with j = p but who support r can feel safe switching to r because they anticipate that, if they were to actually induce the feared p-q-shift, then later voters (we’ve already supposed that there are enough of them) would see the opportunity for a q-r-shift and immediately play the j-r-switch, thus finishing the job. The reason that this is incorrect comes from imperfect information. Recall that our voters play based on estimated “poll numbers”, and not an exact down-to-the-person account of who stands where. Voters would have no way of knowing exactly when the p-q-shift has occurred. In reality — under imperfect information — a voter would fear that the p-q-shift would occur and yet go unrecognized by other voters, who, themselves, would decline the j-r-switch (for assurance reasons; they believe p is still in the lead) and fail to induce the q-r-shift. Then q could win the election.

Note, in particular, that this strategy encourages 2-1-switches (Bush-Walker-Carson) and discourages 2-3-switches (Gore-Bush-Nader). Most importantly, it would also discourage 2-3-1-shifts (Bush-Christie-Carson).

Though the first two are somewhat obvious, the latter should concern us. 2-3-1-shifts are ideal, but they’re impossible to achieve even through an optimal individual strategy.

This brings us to the group game.

The Group Game, and a few loose ends

The players of the group game consist of political organizations which each represent the totality of voters holding some particular utility ranking scheme. In each move, a group must decide whether to extract from the various candidates’ constituencies its respective members, and place them together into a single candidate’s constituency. The situation might look something like this:

An illustration of a group’s decision to perform an A-switch.

An illustration of a group’s decision to perform an A-switch.

Things are much simpler in the group game. Crucially, because a group’s moves are coordinated and instant, a single switch can also induce shifts. (The reason this doesn’t happen in the individual case is imperfect information; see the note above.) There are no intermediate stages, and the only shifts groups can induce are p-shifts and p-q-shifts. It’s now clear, then, how a group should play.

Claim 4: Group play strategy. Consider an election setup with n candidates, 1, 2, …, n, and a particular group G. Without loss of generality, assume that the candidates 1, 2, …, n are named in such a way that candidate i holds the ith position in group G’s utility assignment scheme for all i = 1, …, n. If, for some numbers p > q, G judges that the contingent of qsupporters among all blocks is sufficient to induce a p-q-shift, then G should play a q-switch. If there are multiple such opportunities, G should select the (unique) one for which q is minimal. Otherwise, V should play stay.

The important fact, here, is that where an individual sees a p-q-r-shift, a group sees a p-r-shift. The group can act to instantly overcome collective obstacles. Group behavior coincides with individual behavior in all cases except for the q > p > r case. In this case, the behaviors diverge; individuals encounter assurance games and only groups can resolve them.

This example, then, which began our discussion, has proven to be the star example of this theory. Rather than obscure and complicated, this is the key canonical case in which individual behavior fails to lead to optimal outcome. In this example lies the need for political assurance movements.

Thus, in a way, we’ve finally gotten around to answering the question we asked in the beginning. Why is political organization important? Well, I’m not going to say it again. Look above!

A refinement of both games

To make things more accurate, we introduce a subtle refinement to the theory. Our strategies are too restrictive, in the following sense. It may happen that an individual or group is unable to induce any shifts at all, and our above rules would have that player stay. More generally, the candidates which a player is capable of shifting into victory might represent an inferior subset of that player’s favorites.

At the same time, some candidates can be pushed into victory only through multiple-move sequences. For example, a player might recognize that by contributing to the tally of a distant trailing candidate, he can, not bring him into first place, but rather bring him close enough to first that a future player becomes able to catapult him into first. Even more subtly, in the individual game, a contingent of individuals can act in such a way to convert another player’s 2-3-1 into a 2-1, and to bring that candidate into the lead against all odds. I’ve come up with numerous examples like these. The behavior, indeed, is extraordinarily complex, and I suspect that one could discover move-chains of arbitrary depth. I’ll let the reader explore this if he or she wishes.

The point, then, is this. Players — whether individual or group — should consider not just candidates immediately poised for the presidency, but also those who can arrive there through a sequence of future opponents’ projected rational moves. This can be easily incorporated into the strategy I’ve defined. Indeed, though my definition of shift implicitly incorporates a focus on the leading, first place position — we might even rename them 1st-order shifts — we can also define mth-order shifts, which replace the candidate holding the mth position. We should actually have game players examine all possible shifts of all order in such a way as to favor those which conduce, in the future, to a better candidate’s presidency. I won’t formulate these claims explicitly.

This, though, about completes my theory of electoral strategy. And since you’ve made it this far? Go Ben Carson, 2016.

  1. Wall Street Journal: “Outsider Ben Carson Rises in 2016 GOP Field
  2. New York Times: “How Do You Solve a Problem Like Ben Carson?
  3. This page contains a few links to and information regarding Axelrod’s work on “The Evolution of Cooperation” in iterated prisoner’s dilemmas.
  4. A few definitions concerning assurance games.
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