The Logic of Playing 4 Women


There’s a default in most high-level mixed teams to play 4 men most of the time. The data in this useful article show that quite clearly.

But continuing that default behaviour under the new rules would be foolish. It is logically impossible that both teams are correct to call 4 men most of the time, if their intention is to win the game.

Let’s throw in some made-up numbers.

  • Team A wins this point against Team B 70% of the time when playing 4 men
  • Team A wins this point against Team B 60% of the time when playing 4 women

These are fairly believable numbers. And they suggest that team A is better off playing 4 men.

But these numbers inevitably mean that Team B will win the point

  • 30% of the time playing 4 men
  • 40% of the time playing 4 women.

Team B’s percentage chance of winning this point is always exactly 100 minus team A’s percentage.

p(B)=100-p(A)

There’s only two outcomes possible – you can’t have a draw in this point – and the total probability must add up to 100%. Therefore a rise in one is a fall in the other. A strategy that improves things for one team makes things worse for the other team, and vice versa.

Team B should prefer 4 women whenever Team A should prefer 4 men.

And this is not some fluke of the numbers, but an absolute inevitability. The only situation where one of the teams is not better off playing 4 women is when the chances are exactly the same regardless of 4M or 4W.

Always, and without any exception whatsoever, if one team is better off playing 4 men in this point, then the other team MUST be better off playing 4 women.

I’m not saying that team B will probably win the point if they play 4 women and probably lose if they play 4 men. That could be the case – the numbers could be 60/40 and 40/60 – but even when Team A is favourite in all circumstances, as in the example above, then whichever ratio most favours Team A will always be the ratio for  Team B to avoid.

So, assuming that it’s completely random which team gets to choose 4M or 4W, you should expect precisely 50% of points to be 4 men and 50% to be 4 women. One team is always better off calling 4 women, unless it’s a complete toss up – in which case you’d still mathematically expect 50% of each – so if the right to choose is randomly allocated then we should see both choices equally often.

And – crucially – with the new ‘endzone decides’ rule, it IS completely random (over the long haul) which of Team A and Team B has the choice of ratio.

Over a large number of games and teams, at the highest level, you should expect to see precisely 50% of points played with 4 men and with 4 women.


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On any given point, the optimal choice of 4M or 4W is likely to be determined by some combination of the following:

  • Relative strength of [our men v their men] and [our women v their women].
    • It may be that one team is better across both genders, but still the gap will normally be larger with one gender than the other¹. It’s also possible that one team is just better at strategising for one ratio or the other – so, it isn’t completely about the strength of players, but more generally about whether you are relatively more competitive against this team as 3:4 or 4:3.
  • Offence versus defence.
    • It may be that one ratio is more suited to scoring clean O possessions versus disrupting the opponent, regardless of the actual strength of the players.
  • Upwind versus downwind.
    • It may be that one ratio is more effective in one direction, regardless of O/D or player strength.
  • How tired your men/women are versus how tired their men/women are.
    • Teams often choose 4M simply because they’re short of women, which can make sense if you brought an unbalanced roster and your aim is to protect your players from exhaustion. But in terms of winning this game, your energy levels don’t matter – it’s whether you are more or less exhausted than the opponent. Even if both teams are short of female-identifying players, it remains the case that one of them will have a competitive advantage in calling 4 women, since the other team is even more tired.

But having said all that, it doesn’t matter why one team prefers 4M or 4W. It doesn’t matter how the above reasons – or any others – combine to give a percentage chance of scoring for any particular point. There is no additional argument needed about the whys and wherefores, because of the simple maths above. If A is correct to prefer 4M, for whatever reason, then B must be correct to prefer 4W.

The second part of the overall argument is about the new ‘endzone decides’ rules. Under these rules, there is no particular reason that either offence/defence, or upwind/downwind, should have the choice more often than the opponent, across enough games.

You might find that a game has a run of holds, so that for example the team on offence (or the team going downwind) consistently gets to choose the ratio, and that could mean that in any particular game you might be nowhere near a 50% balance of 4M and 4W points. But across enough games, that will average out. Given that there’s a separate toss for who chooses ‘Endzone A’, it is random chance whether the upwind or downwind team, or the offence or defence, chooses the starting ‘Endzone A’, and so you’d expect no particular inherent bias².

Across enough games, you’d expect each ratio to be chosen 50% of the time, on average, if both teams were solely looking to maximise their chances of winning.


It seems to me there are four reasons why, even at the highest level, the default choice is 4 men.

  • ‘Our men are better than our women …’
  • ‘We have more men than women …’
  • Years of practice in ‘4 men’ ultimate
  • ‘Offence chooses’

Many teams feel that their male players are simply better than their female players and that therefore the best team they can put out is the one involving more men. It isn’t required that the men are actually stronger players than the women for people to think this way. All that is required is the belief that the men on this team are better players, and that belief is unarguably very prevalent on most mixed teams.

In practice, even ignoring any biological differences, you can make the case that the majority of elite male mixed players will be stronger than their female counterparts based purely on the membership demographics in most areas. The 12th best of 1,000 local male players is likely to be a lot better than the 12th best of only 100 local female players.

But even if you really are a ‘better’ team with 4 men on the field, this disregards the fact that by calling 4 men you are also giving the opponent 4 men. Even if it’s true that your men are better than your women in this particular game, that doesn’t automatically mean that your men will compare more favourably with your male opponents than your women do with their female opponents.

The strongest team you can put out is not always your best option if it also plays to your opponent’s strengths. That couldn’t always be the case, as we discussed above.

A second possible reason for the default 4M choice is that you didn’t bring enough women to the event, and hence you don’t want to exhaust them. This is likely related to the issue above, about a perceived drop in standards for your 12th-best woman that you don’t see in your 12th-best man, which biases your roster towards men³. But unless you believe you’re the only team facing these kind of selection pressures, then you’ve made an error. It matters not a jot how good your 12th woman is compared to your 12th man, but compared to the opponent’s 12th woman. If you want to win games, that’s the comparison that matters.

If you find yourself, in the bracket at Nationals, calling 4 men simply because you have more men, then you should have brought more women. At lower levels you may have no choice; at the very top level, you certainly do. If all the other teams are consistently rostering way more men than women, you have a strong competitive advantage in taking an even roster and making their individual women play more points. They can’t try to tire you out by forcing you to play 4M on every point; but you can always choose to make 50% of the points 4W.

Another possible reason for a male bias in ratios is simply due to the experiences players have had at lower levels, where a shortage of female-identifying players might lead to a lot of 4 men points. Thus the developing players get a lot of practice in, and spend a lot of time strategising for, 4M mixed ultimate. It’s easy to see how that can carry over to higher levels where recruitment is less of an issue. But you can also easily see the competitive advantage in playing 4W against such a team.

And the last reason is the old rule about ‘offence chooses’. If it’s the case that offence/defence makes a difference to the optimal choice of ratio, than it’s no surprise that people who grew up in that ‘offence chooses’ system would have a bias one way or the other. But any remaining bias from that is now out of date, and once again, there is a competitive advantage in a more balanced approach. If it is true that offence is easier with 4 men, and you’re prepared to call 4 women on defence while they are not, you will win more points than you otherwise would.


Overall, in a rational world of perfect information, under the new rules, 4-women points should be called precisely 50% of the time (on average across many games) if all teams are serious about winning each game. Anything else shows a bias unsupported by competitive reality.

Of course, we don’t have perfect information. We don’t necessarily know that our chances with 4 women are better than our chances with 4 men (against this particular opponent, or in this particular upwind point, or whatever). How does that change things?

It doesn’t.

Even if we’re just guessing, there’s no reason to assume that 4 men is more likely to be our best option. It CANNOT be the case that 4 men is better for both teams on any given point, and therefore it is not rational to assume without information that 4 men is a better choice for you than 4 women. If you have no information, it’s a coin toss, and we’d expect – yet again – an even split between 4M and 4W. Teams with an open mind will on average perform better than those who start by assuming that 4M will win the game (and take a few points to talk themselves out of it).

Many high-level mixed team captains/coaches are not making rational decisions⁴ if the ratio between 4M and 4W does not approach 50%.

QED.


¹ Unless they are better by precisely the same amount across both men and women, in which case it’s a toss up and we still wouldn’t expect any deviation from an overall 50% 4M and 50% 4W.
² At least, if we include WFDF’s stipulation that the ‘choosing endzone’ changes at half time. It’s possible I guess that in the US, where there is no half time switch, teams will have a consistent preference for picking the upwind or downwind endzone as Endzone A at the start of the game, and that could skew the overall stats slightly.
³ At lower levels, it’s very reasonable that winning isn’t your only goal. You might no want to exhaust or even injure your few women, even if the other team were even more exhausted and you could win the game that way. It probably isn’t your fault that your team is unbalanced. But at a high level, you should have chosen a roster that allowed you to make the best possible competitive choices.
⁴ Well, there’s another option. Perhaps they’re rationally reacting to an irrational team. It might be rational for a captain to follow the status quo if calling 4 women will cause the men on the team to become angry, unfocused, or confused, and thereby play less well. But for sure, someone somewhere is being irrational if the long term average isn’t somewhere near 50% 4M and 4W.
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9 Responses to The Logic of Playing 4 Women

  1. Hildo Bijl says:

    I completely agree with your main point. I really like how this is brought up, because it totally demolishes the argument of some coaches to mainly field four men. There’s two follow-up conclusions that are not correct though, and I hope that, by pointing them out, I can make the argument even stronger..

    The first one is about, when you’re short on women, it’s irrational to “save” the energy of your women. If you are short on women, and your opponent is even more short on women, then for THIS game it’s wise to put in 4 women more often than not. After all, you’ll gain a relative advantage over your opponent. However, the result is that, after this match, the women of both teams will be totally spent. Then, in the next match, you’ll face a new team with fresh women, and you’ll end up going down hard. So throughout a tournament (with the exception of the last match), it can be wise to save the energy of the women (or the gender you’re short on), even if the opponent has more of a shortage for that gender.

    The second incorrect claim is that, if everyone is fully rational, then eventually we should theoretically see a 50%/50% ratio in 4M/4W. To see why this is not true, imagine that we exaggerate the situation. Suppose that fielding 4M as offense gives you a 100% to score, while fielding 4W gives you a 0% chance to score. (I know, it’s not realistic, but even if the numbers are 77% and 76%, over the long run a trend will be visible.) Now there’s two options:

    1. The offensive team is on the “choosing endzone” and gets to decide whether to play 4M or 4W. Naturally they’ll choose 4M because they’ll have a 100% chance to score. The next point the other team is offense andthey’ll be in the choosing endzone, so the next point we’re in this situation (1) again.

    2. The defensive team is on the “choosing endzone”. They of course decide to play 4W, because this gives them a 100% chance to score. They’ll do so and score. The next point, they’ll still be defense, and the other team will still be on offense, but now the other team is in the choosing endzone. That means that, in the next point, we’re also in situation (1).

    The conclusion: in this situation we’ll wind up with a 4M choice at every single point, apart from perhaps the first point. It’s not necessarily a 50%/50% ratio. Of course, in a real Ultimate match, this will not be the case. But even if 4M scores their offenses 80% of the time, and 4W scores their offenses 75% of the time (percentages which I think are realistic for reasonably high levels of Ultimate) then this effect could be visible.

    Other than that, a really good article and I still wholeheartedly agree with the main point.

    Like

    • Thanks Hildo. Really interesting. The first bit I had considered but didn’t want to get into as it was complicated enough. The second point is very clever, and I think you’re right. It might not be that serious in practice, where we have upwind/downwind to worry about also, and it will depend how big the advantage of one ratio or the other on offence would be, but I think you’re right that it would be expected to skew the results slightly. All hail the first statistician to put numbers on that one!

      Like

  2. Steve says:

    This assumes, however, that winning the next point is the one and only concern. In a tournament setting you often have other concerns, such as making sure players get sufficient rest so they’ll be able to go in the next game.

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    • Absolutely. But if you believe there’s a competitive advantage in playing 4 women then you should roster more women. Perhaps not 50/50, if you think the opponent will always call 4 men and you will often but not always call 4 women. But certainly closer to 50/50 than your rivals.

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  3. One way that I think the current imbalance is maintained is that large numbers of mixed captains think that their opponents are making a strategic error by not playing 4 women.

    Also, many of these decisions are driven by factors other than pure desire to win the particular game at issue. Here are a few examples:

    Nationals team A is playing Regionals team B, where A would be significantly benefited by playing 4 women the whole time. But A is going to win anyway, and playing 4W would tire out B’s women/be seen as running up the score/tire A’s women more since there are more men than women on A/etc.

    Team C has 15 men and 9 women. Playing 4W would be beneficial for C in winning the current game, but the women don’t want to play totally exhausted/are newer to the team and/or ultimate and thus need to be retained/etc.

    Team D has 15 men and 10 women. The men complain about playing 4W because then they don’t get enough playing time.

    Team E and F are playing, and the game isn’t very significant so people just choose what they’re used to, which is 4M.

    Many of these attitudes rely on 4M being the “default”, but that’s likely to be that way at less competitive levels for a long time.

    Like

    • Yep. But all those attitudes also rely on having a severely unbalanced roster, which (for top teams) is very fixable if they buy into the idea that 4 women will win them more points.

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      • First, I think there are maybe 20-30 teams without an unbalanced roster in the USAU mixed club series (I don’t think this includes all Nationals teams, for example). Second, almost everyone spends lots of time playing at unbalanced levels, leading to these habits.

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        • Oh yeah, absolutely, wasn’t disagreeing with you. Rather, I was suggesting that high-level captains reading the article might want to think about the competitive advantage of moving their roster closer to balance, IF they are suffering the issues you mention.

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  4. mkt42 says:

    If anyone reading this article has taken an economics class, they should recognize that this is an example of the importance of “comparative advantage” (and the unimportance of “absolute advantage”, i.e. which team is just plain stronger).

    The article correctly elucidates the underlying principles, but I also agree with Hildo Bijl’s comments about how the analysis needs to be modified.

    Like

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