Braess's Paradox: Mathematical proof that having more options can be a detriment, even in a situation where everybody is perfectly rational and has perfect information.

So we've been getting posts about whether people having more potential options in the dating marketplace is good or bad and there were a surprising number of posts arguing stuff like "the only reason that more options can be bad is poor decision making or choice paralysis (the person is not perfectly rational)" or "choice gives the illusion that there is someone better out there and so makes people pass up on perfectly good partners (imperfect information on the true state of the market)".

While these are both perfectly valid arguments showing why more options aren't always good I'd like to add to that with an example of how more options can be bad even when everyone is perfectly rational and has perfect information. In fact this also is an example showing how adding new options can make life worse for everyone while also forcing them to take this option unless they want even worse outcomes (i.e. there is no "just don't take this option if you don't like it"; the simple existence of that option is enough to ruin the previous better choices). This is known as Braess's Paradox if you want to read up more about it:

https://imgur.com/a/Ams3IoZ

Consider a case where 4000 truck drivers want to go from the Start point to the End point as quickly as possible. They have two routes available to them: going via A or going via B (ignore the dashed line for now). If they go via A they take the road from start to A and then from A to End. If they go via B then they take the road from Start to B and then B to end. The time to travel each road is given by the number on the arrows representing the road. T is the total amount of trucks that take the given road, so e.g. if 2000 trucks went on that start to A road it would take all of them (1000/100 = 20) minutes each. The Start to B and A to end roads always take 45 minutes to travel regardless of how many trucks use them).

NB: in real life the first case corresponds to a well developed road where what limits the speed of the trucks is congestion, the second case (constant 45 mins) models a rocky, bad road where the terrain is what limits truck speed.

Since regardless of whether a truck goes via A or B it will have to travel one T/100 mins road and one 45mins road the optimal result is when 2000 trucks go via A and 2000 trucks go via B (because for any other split a truck on the more busy line could switch to the less busy one and save time). This leads to a travel time of 45+(2000/100) = 65 minutes each.

Now suppose we add in another option where we allow instant travel between A and B (the dashed line in the linked diagram). This basically collapses A and B into a single node where we have two roads each from Start to the node and from the node to the end, where one road takes 45 mins and one takes T/100 mins.

Now the calculus is changed, since the maximum time the T/100 road can ever take is 4000/100 = 40 < 45 there is no reason to ever take the 45 minute road for any driver. So they all take the T/100 roads which in the original diagram corresponds to all 4000 drivers taking the T/100 road to A, taking the free road from A to B and then taking the T/100 road to end. This means each driver is now spending (4000/100)+(4000/100) = 80 minutes to get from start to end, which is a whole 15 minutes longer than before "free" route between A and B was added.

Another way to see this is to look at what happens if we still had the old distribution of 2000 bottom. Now a driver going top could notice that if instead of going A to End (45 mins) he took the free road A to B and then went B to end which would take him (20+2001/100) = 40.01 minutes, saving him almost 25 minutes. Hence he will make this switch making the B to End road slightly more congested.

Similarly any driver taking the bottom route notices that if instead he took the top route and the free road from A to B he would spend 2001/100 = 20.01 minutes to get from Start to B rather than 45 minutes, so he would switch too, making the Start to A road slightly more busy. Repeat this line of thought for all drivers making the rationally best choice for themselves at all times and you end up with all traffic going Start to A; A to B; B to End and taking 4000/100+4000/100 = 80 minutes.

Hence we see how adding extra options has made life literally worse for everyone involved, even though they all took perfectly rational decisions at all times and had perfect information. Note that drivers can't even choose to go back to the old paths, since now the Start to A; A to End route takes (4000/100)+45 = 85 minutes which is even worse! Hence they are all forced to use this new "optional" route between A and B which has just served to make life worse for the drivers as well as made their old paths unviable.

The correct way to fix the issue here is not education or anything, the drivers have perfect information and know exactly what is happening here. They know that if nobody took the free route at all they could go back to 65 minute travel times, but if they did that then there would be an incentive to defect and take the free route (cutting your time to 40.01 minutes while making it slightly worse for everyone else) which makes it an impossible to solve coordination problem.

The correct solution is to take a jackhammer to the road between A and B and destroy it; that way improving the time for literally everyone. This "paradox" is actually pretty common in real world transport networks; indeed NYC pedestrianised Times Square for exactly this reason in 2009 which actually reduced congestion and made car journeys through the surrounding area faster.

This also applies to dating markets. Here the "free route" option is represented by online dating and all its introduction does is make life worse for almost everyone as well as force people into using it, because the very fact it exists makes previously good methods unviable to the point of being worse than using it (just like how the free road from A to B made the Start to A; A to End route unviable). The solution here is again to severely restrict OLD and make it much much harder to use (i.e. make it so that its more difficult to sign up and use an app to find a partner compared to e.g. going to a bar, this would be like making the A to B route take 30 mins instead of being free, which would remove the paradox) that way fixing a lot of our societal problems related to dating.

Discuss.