Michael Eriksson's Blog

A Swede in Germany

Oddly equal elections and game theory (and some other thoughts)

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With the U.S. election day upon us, I am pondering the oddity that elections tend to be close or closish, no matter the parties/candidates and positions. (And, no, e.g. 55–45 is not that much. That it appears to be a landslide shows the scope of the issue. If numbers had been random, for instance, it might equally have been 95–5.) A portion of that is certainly that most voters are stupid, uninformed, and/or tend to believe what they are told (possibly, with a reservations for who does the telling); and, certainly, other voter centric causes might be relevant. However, right now, my thoughts are on game theory, that the parties might deliberately position themselves in an attempt to gain a majority, and thereby are forced to take positions more or less close to what the median voters want to see—and this would neatly split the field in a manner similar to what is observed in actual politics.

Consider first an extremely simplified example with two parties (resp. candidates), a single issue with a simple numerical indicator where the parties can make a “bid”,* with voters who pick the party whose bid is the closest to their respective preference, where the parties have perfect knowledge of voter preferences, where winning is all that matters, where both parties play perfectly rationally, and where there are no hidden side-effects. Further, strictly for the sake of easier-to-understand examples, assume that the numerical indicator is strictly integral, implying e.g. that “5” is a valid bid, but “5.5” is not, and that the median** used below is also integral.

*E.g. in that one party might bid a something-or-other of “10” and another of “20” in some unit, but not “more money for schools; less for tanks”.

**The median of 6 and 7 is 6.5, which is not a valid bid and would cause problems.

Let us say that the bid “5” catches the median voter. If the one party picks “5”, and the other something else, say, “9”, then “5” will almost always win and never do worse than a tie: Per definition, at least half of all voters will be at the median value or smaller and at least half will be at this value or larger.* Already here, the median is guaranteed to have at least 50 percent of the vote, regardless of whether the competing bid is higher or lower. However, in reality it is even better: assume a real-valued set** of values with a median of m and an x > m*** with half (or more) of all members >= x. By the definition of m, we also have (at least) half of all members <= m. If the “or more” holds, at least one member must necessarily simultaneously be both > m and <= m (and >= x, < x), which is impossible. If it does not hold, there is a small loophole, in that exactly half of the members can be >= x and > m, while exactly half are < x and <= m. This, however, presupposes that m is not equal to any member (as with e.g. the set 1, 9, with the non-member median 5, where e.g. x=7 achieves a tie). If m is a member, in the aforementioned constellation, it is not the median, per definition. (What about voters between 5 and 9, using the original numbers? They do not matter, as 9 is not the above x. With the rules specified, voters below 7 go with 5 and voters above 7 go with 9, while voters at 7 are tied. But let us give even the voters at 7 to 9. We now have the following observations: (a) The number of votes for < 7 is >= the number of votes <= 5. (b) Per the above, excepting that one small loophole, even the number of votes <= 5 (m) is strictly larger than the number of votes >= 7 (x). (c) A fortiori, the number of votes awarded to 5 is strictly larger than the number of votes awarded to 9. If the loophole hits, we have still have no worse than a tie.) Correspondingly, the best strategy for both parties is to pick the median value, resulting in a tie.**** As both parties have perfect knowledge and play perfectly rationally, this is what they will do.

*If this seems paradoxical, consider the set 1, 2, 2, 3, with median 2, and where there are three entries (1, 2, 2) <= 2 and three entries (2, 3, 3) >= 2.

**Re-writing this paragraph, past 4 AM German time, to make it understandable, I imagine that a set must have unique members. If so, here and elsewhere, please make a corresponding mental adjustment to allow repetitions, as with 1, 2, 2, 3 in the previous footnote. As to the text as a whole, I (a) suspect that I am making matters worse, (b) would, had it not been for the readers, prefer to make it more mathematical, including more formal notation and replacing the median with some type of equivalence class of “medianoids” where all numbers between the largest number smaller than a non-member median and the smallest number larger than it are considered equivalent. (For member medians, the median would be the sole “medianoids”.)

***m < x is handled analogously.

****What happens now is not specified above, but we can e.g. imagine that voters distribute themselves randomly between the positions “first party”, “second party”, and “bugger this voting nonsense—I’ll stay at home”. In this case, the difference between the vote counts of the two parties will be close to 50–50 (of those who actually vote) and statistical fluctuations will determine the winner and the margin of victory.

Unless the number of voters is very small,* we will now expect an at least approximate draw (cf. the above footnote), or the type of close to 50–50 numbers that we see in real life—in fact, likely, numbers far closer to 50–50 than in real life.

*With a single voter, one party will win 100–0, while e.g. a twenty voter group might occasionally land at 75–25 through coincidence. In contrast, for a group of twenty thousand voters a 75–25 would be a sign of some systematic difference, contrary to above assumptions.

A somewhat similar image will appear if we loosen constraints. For instance, if remove the “perfect knowledge” restriction, the parties must guess what the optimal bid is. With a one-off election in an unknown territory, this is a fool’s game, but both parties have a 50–50 chance of winning or drawing (even be the eventual distribution of voters far from 50–50). If they guess somewhat similarly, they will likely have numbers close to 50–50, and chances are that they will do so, given the chance.* If in doubt, going for the same bid will give that tie or almost-tie. In the real world, the game of voting will repeat every few years or, in different areas and for different positions, more often, and great effort is spent on probing the population, implying that a reasonably good guess for the perfect bid might be available.

*This will depend a little on the modalities, but take a setting similar to a prolonged campaign where bids can be adjusted over time. On day one, the parties bid 10 and 20, respectively. Regardless of who is closer to the truth, the first party is now giving up the range 15–19 (15 is equally far from 10 and 20 and they draw there), and will re-bid “19” to take that range, while the second gives up the range 11–15, and will re-bid “11”. After a few rounds of jumping back and forth, they will land at 14 resp. 16, and might end it there, both bid 15, try to bid in some other constellation, whatnot, but ultimately in a pattern that results in a small or no difference, as to not gift a range to the opponent. (There might be some deviations if one or both take the strategy of the other into account, but not normally any that change the principle.)

Or say that we have more than one issue to bid on: The voters will use some implicit* aggregate function for judging the sum of all issues, and we still have a win/draw strategy if we can pick a combination of bids that matches the median.** More importantly, in any somewhat reasonable scenario, over- or under-bidding the opponent in the right direction will lead to the right result, just like in the above footnote, which will bring both parties to manoeuvre into approximate 50–50 scenarios—if in doubt, if nothing else helps, by just matching the exact bid of the opponent. (As above, if we still had perfect knowledge for both parties, the rational strategy for both is to use the same set of bids, except that there might be some room for different sets of bids with the same number of voters. There could conceivably be constellations where a better bid can be found, but the effect will be extremely temporary before the opponent retaliates or matches that bid for a tie.***)

*Not necessarily a mathematical, rational, consistent, or whatnot function, but likely with what amounts to one function per voter.

**There is some risk that we can not; maybe even, in reverse, that there might be more than one combination of “median bids” conceivable. I have not done the math.

***I have not done the math here either.

What if winning is not all that matters?* What if, e.g., one party has an upper limit of “15” for whatever bids it can make in good conscience or wants to go as low as possible while in office? Well, this makes things tricky, especially if we only have one issue to bid on. However, even here a similar type of over-/under-bidding might tend to close the distance.** From another angle, politicians are not known to be truthful, and a dishonest bid of, say, “19” might still follow in the above example, with the idea that the actual policy, once in power, will be no more than 15, regardless of what bids were given. From yet another, many view it as better to gain or remain in power, even with sub-optimal politics, because the alternative might be (or be believed to be) worse, with the result that “19” is bid anyway, because a policy of 19 might be worse than one of 15—but better than one of 20, which was the opponents first bid.

*How often this applies in real life, for what candidates, and for what parties is a very, very interesting question.

**For instance, take the last example with an added upper limit of “15” for the first party: The bids “10” and “20” might now be followed by “15” and “11”, then “10” and “14”, “13” and “11”, “10” and “12”, and a new halt with close values. (The exact sequence will depend on priorities, but this illustrates the principle.)

Remove the perfectly rational play, and things might again grow more interesting, but going just a little higher or lower than the opponent is not rocket science, both parties will (all other things equal, on average/when looking at expectation values) make mistakes of a similar number and magnitude, and chances are that things even out.

The last brings us to an advantage of having many unrelated issues: a party that misjudges one or two issues severely, or many a little bit, might well see these misjudgments neutralized on average, because they are put on the scales against mistakes of the other party and, from another point of view, against own excellence on other scales. Have, say, twenty issues that at least some voters are interested in, and a distribution close to 10–10 in issues “won” and “lost” with the voters is statistically likely, and might lead to numbers close to 50–50 in terms of voter percentages. This even as we remove constraints and make the scenarios more realistic.

The topic could be pursued in much greater depth, but I will settle for some additional remarks (mostly the result of free association while I was writing the above):

  1. In some situations it might pay to game the opponent. Assume e.g. that one party cares less about winning the election and more about what policy is implemented post-election. Then there might be room to go artificially low or high, and depend on the opponent to stay close. Take, again, starting bids of “10” and “20”. The first party wants as small a value as possible, and might now choose to not change its bid, leading to “10” and “11” in the next round, or might choose to lower the bid, e.g. with a next round of “5” and “11”, a third round of “0” and “6”, etc. until a further lowering is either not credible or not sensible. This might well lose the election, but the policy of even “11” (let alone “6” or lower) might be sufficiently much better than the final own bid of “14” from the original example as to outweigh this. (And, yes, I suspect that similar things happen in real politics.)
  2. Contrary to what would be expected from the above, the distance on many issues can seem quite large, even to the level of polar opposition. (Abortion in the U.S. springs to mind.) This does, in part, speak against the idea, but might partially be an effect of the sheer number of issues available, where it might be less a matter of winning individual issues and more of winning in aggregate (cf. the statistical argument above). Other aspects include that various issues might not be unrelated, which can skew impressions, and that the difference on some issues might seem greater than it is, because the opinions are not at extreme ends of a free spectrum, but of one constrained by opinion corridors, Overton windows, or whatnot. In some cases, these can be narrow indeed. (I note e.g. that even the allegedly non-Leftist main parties in Sweden are onboard with various Gender-Feminist nonsense as core beliefs, excepting only SD—which is, unsurprisingly, borderline untouchable to the others.*) It might also be that the many individual voters would miss differences in opinion if they are too small (or that the parties believe that voters would miss differences), which might lead to surprising result from the point of view of a more discerning observer.**

    *Here we see another potential distorter: if a non-pariah party comes too close to a pariah party on some even semi-controversial issue, there is a risk of guilt by association, of being (fairly or unfairly) grouped with the pariah, etc., which can give strong incentives to limit the range of real or metaphorical bids. Indeed, looking specifically at migration in many countries, this is exactly what has happened.

    **To return to our “10” and “20” initial bid: if the parties (a) assume that a voter might miss a difference of 1, 2, or maybe more, (b) want to ensure that they never, ever cross lines (unlike the first version of the example), we might end up with a bid series of “11” and “19”, “12” and “18”, “12” and “17”, “11” and “17”, or similar.

    As to the reasons that the gaps are not shrunk for purposes of vote fishing, I suspect a mixture of irrationality in strategy, the sheer extremeness of many Leftist opinions, which might make a “follow the leader” game intolerable to both sides, and a propaganda strategy, especially on the Left, to not allow the possibility that the opponent has any good points, that anything the opponent believes is automatically wrong, etc. (I have written about such behaviors on the Left in the past.) This might also play in to widen the gap, because condemning someone as “evil” for bidding “10” is harder when the own bid is “15” than when it is “50”. It might also be that politicians are highly principled, but, well, let us be realistic here.

  3. Credibility might often be an issue, and might play in with the previous item: if two parties are too close to each other, let alone skips over another in terms of who is “highest” and “lowest” on some issue, this can lead to credibility problems. (Unless these are comparable parties and allies in a multiparty system.) However, looking at sufficiently long-term perspectives, this changes to some degree. It can, for instance, be argued that the U.S. Republicans and Democrats have changed places on some issues, like working-class life and attitude to Big Business. There are also cases where the distance between two parties has been kept approximately constant, while both parties have drifted in the same direction, maybe because the one party is trying to get closer and the other tries to keep the distance. (Likely self-defeating, as this might also shift opinion corridors without altering voter shares—or maybe even causes massive voter dissatisfaction.) Potential examples of this include climate issues in a great many countries and Gender-Feminism in at least Sweden. Another potential example is the drift towards Big Government, but this might have other explanations, maybe in the “slippery slope” or “boiling frog” families.
  4. If there are not enough issues, or not enough issues with enough room to profile oneself relative others, creating issues is an option. Here I would point to many suspect Leftist issues and sub-issues. Consider e.g. the historical record of environmental doomsday prophecies that have not panned out, or the unreasonable and, in its effects, anti-environmental hatred of nuclear power pushed by many “Green” parties; or the claims around “Systemic Racism” and “Patriarchy” that only seem to be there due to naive or deliberate misinterpretation of facts, statistics, and causalities. Biden’s recent hate propaganda of “the others are fascist (and don’t look too closely at what I do)” is likely another example. (However, determining what is malice and what incompetence can be hard.)
  5. The extremely one-sided election results seen in e,g. some Communist dictatorships with more-or-less nominal voting, where the Party and Comrade X are re-elected by 99–1, are an extremely strong sign of direct or indirect cheating in my eyes and partly based on the above. This is unlikely to be a great surprise to the reader, but much might be explained simply by persistent and one-sided propaganda, news reporting, whatnot, absent the general drift towards 50–50 results. A result like 99–1 simply goes beyond what is plausible based on just natural opinions in all but the most extreme cases. More likely, it is supported by means like voter intimidation and non-secret ballots, outright manipulation of results, that dissenters do not bother to even go to vote, and similar. (The exact means might vary strongly from country to country.)
  6. The general reasoning applied to a two-party system above can also be applied to a multiparty system, but with the “advantage” that the parties of a certain block need not be good at everything individually (it might be enough that one of them has a strong grip on a particular “market”), and the complication that they must not get too much in the way of each other. (In particular, it can be that even a good bid from the two-party system is no longer a good bid, because it is cut off on both sides. There is, in particular, no guarantee that a median bid will win. Further, matching someone else’s bid will tie that someone, but not everyone else, and the sharing of votes can harm both relative the other parties.)

    However, an interesting further driver of 50–50 situations is that (in particular) smaller parties might be tempted to switch blocks or go more independent if the old block grows too strong. For instance, a 10-percent party might be better off helping a 41-percent party to power than a 49-percent party, because it might get a greater say in exchange. For instance, in Sweden, a position as “vågmästare”* has been historically attractive, as it carries a lot of influence but little responsibility. Similarly, a small party within one block might well conclude that it can grow larger by changing its politics, be it as an independent or in another block.**

    *Roughly, “master of the [weighing] scales/balance/whatnot”, the one who can put a little extra weight on the one side to push the one scale down at the cost of the other. This refers to a party which is not member of a fix block, or in government, but can play two large blocks against each other and get favors from both in exchange for votes in the “right” direction on important issues.

    **This, I suspect, backfires more often than not, but hope springs eternal.

  7. The “advantage” of having multiple issues available can be severely limited by an apparent drive to have every party member agree on everything (especially, on the Left). Too often, there is an attitude that “either you agree with me on everything or you are evil [not one of us, whatnot]”. This can not only hamper individuals but also cause a long-term lock-in or a drift towards ever more extreme attitudes as the individuals try to out do each other in orthodoxy.
  8. An optimal use of multiple issues can be hindered by a need to compromise internally, with external partners, and with voters. Here we can also see cases of bartering. Say, e.g., that someone has two issues to bid on and wants to bid “25” on the one and “49” on the other. It might, however, be that going beyond “40” on the second measure will scare away or antagonize someone important, or that “49” is acceptable if the bid on the first issue is lowered to “15”.
  9. Multiple stages of voting, like in the U.S. (with primaries and regular elections) can lead to distortions in tactics. For instance, it might be that a bid of “15” would have been ideal in the regular election, but that a wish to win a primary forced an earlier bid of “5”. Going from “5” to “15” might or might not be acceptable in a next step, but often there will be a lock-in effect or a need to compromise with, say, an “8” as the closest to “15” that will be tolerated by those once swayed by the “5”.
  10. A more realistic model family than the one used above would consider effects on allies. Assume e.g. that we have several individual candidates from the same party running for different offices at the same or close to the same time. To have them run on too different platforms might be very harmful, especially with credibility among voters, consistency of message, and ensuring that the right message has been received. This can then severely limit what bids are available. For instance, it might be that a bid of “100” on some issue would strongly help a local candidate in one state, but also that his party fellows in other states have bid “20” on the same issue. He might now have to moderate himself to, say, “25” or “30” for a much smaller help.

Written by michaeleriksson

November 9, 2022 at 4:40 am

One Response

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  1. […] recent in-the-middle-of-the-night attempts to explain math ([1]) have left me with a bad conscience. I will give myself a do-over with a clearer head and try to be […]

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