Rational Suckers

Braess’ paradox, a multiplayer Prisoner’s Dilemma, leading to avoidable suffering

Why do people skip queues, cause traffic jams, and create delays for everyone? Who are these misbehaving creatures lacking basic cooperation skills? Are they really all that different from you? Are you perhaps one of them?

Various situations involving social interaction drag you into a negative sum game, and make you part of a misbehaving gang. Welcome to Braess’ paradox.

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Each morning at rush hour a total of 600 commuters drive their cars from point A to point B. All drivers are rational individuals eager to minimize their own travel time. The road sections AD, CB and CD are so capacious that the travel time on them is independent of the number of cars. The sections AD, and CB always take 10 minutes, and the short stretch CD takes no more than 3 minutes. The bridges, however, cause real bottlenecks, and the time it taken to traverse AC or DB varies in proportion to the number of cars taking that route. If N is the number of cars passing a bridge at rush hour, then the time to cross the section with this bridge is N/100 minutes.
Given all these figures, each morning each individual driver decides which route to take from A to B. Despite the freedom of choice for each commuter and despite all traffic flow information being available to each and every commuter, the outcome of all individual deliberations creates a repetitive treadmill. Each morning all 600 commuters crowd the route ACDB and patiently await the traffic jam at both bridges to resolve. The net result is a total travel time of 600/100 + 3 + 600/100 = 15 minutes for each of them.

Does this make sense?

At this stage you may want to pause and consider the route options. If you would be one of the 600 commuters, would you join the 599 others in following route ACDB?

Of course you would. There is no faster route. Alternative routes like ACB or ADB would take you 600/100 + 10 = 16 minutes, a full minute longer than the preferred route ACDB. So each morning you and 599 other commuters travel along route ACDB and patiently queue up at both bridges.

One day it is announced that the next morning the road stretch CD will be closed for maintenance work. This announcement is the talk of the day. Everyone agrees that this planned closure will create havoc. Would section AD or CB be closed, it would have no impact as these are idle roads. But section CD is used by each and every commuter. What a poorly planned maintenance, a closure of such a busy section should never be scheduled for rush hour!

The next morning all 600 commuters enter their cars expecting the worst. Each of them selects between the equivalent routes ACB and ADB. The result is that the 600 cars split roughly 50:50 over both routes, and that both bridges carry some 300 cars. Much to everyone’s surprise all cars reach point B in no more than 300/100 + 10 = 13 minutes. Two minutes faster than the route ACDB preferred by all drivers.

How can this be? If a group of rational individuals each optimize their own results, how can they all be better off when their individual choices are being restricted? How can it be that people knowingly make choices that can be predicted to lead to outcomes that are bad for everyone?

Asking these questions is admitting to the wishful thinking that competitive optimization should lead to an optimum outcome. Such is not the case, when multiple individuals compete for an optimal outcome, the overal result is an equilibrium and not to an optimum. We saw this in the game Prisoner’s Dilemma, and we see it here in a situation referred to as Braess’ paradox.

A question to test your understanding of the situation: what do you think will happen the next day when section CD is open again? Would all players avoid the section CD and stick to the 50:50 split over routes ACB and ADB, a choice better for all of them?

If all others would do that, that would be great news for you. It would give you the opportunity to follow route ACDB and arrive at B in a record time of about 9 minutes (300/100 + 3 + 301/100 minutes to be precise). But of course all other commuters will reason the same. So you will find yourself with 599 others again spending 15 minutes on the route ACDB. And even with the benefit of hindsight none of you will regret the choice you made: any other route would have taken you longer. Yet all of you surely hope that damn shortcut between C and D to get closed again.

And don’t assume this phenomenon doesn’t occur in real life.

Homo Retaliens

Why the urge to retaliate?

Game theory models human decisions based on just two characteristics: rationality and selfishness. This minimalistic approach teaches us a lot about the character of economic behavior and the emergence of strategies built on cooperation, competition, retaliation, etc.

By far the most well-known game-theoretical scenario concerns the prisoner’s dilemma (PD). This game is rather boring from a game theory perspective, yet over the years it has attracted an impressive amount of attention, particularly so in the pop science media. The reason for all the attention is that the predicted outcome for this simple game surprises most people. Game theory tells us that in PD rational players focused on optimising their own gains will knowingly avoid the win-win situation for this game and settle for a smaller return.

How can this be?

The simple answer is that in any game rational players will end up in an equilibrium of individual choices, and not necessarily in an optimum. PD is a game designed to render manifest the difference between the equilibrium outcome and the optimal outcome.

An example PD scenario runs as follows: you and your co-player are facing a task. Both of you have the choice to contribute towards accomplishing the task or to free ride. You both have to decide simultaneously and without any opportunity for communication. If both of you decide to free ride, nothing gets accomplished and you both walk out with no gain. Each player putting in an effort increases the total gain by 4 units. This gain will be split equally over both participants. Putting in an effort comes at an individual cost of 3 units.


It should be clear that this game forces a rational selfish individual to free ride. Regardless what choice the other makes, avoiding any effort form your side makes you avoid an investment of 3 units that would deliver you only 2 units gain. As this applies to both players, the outcome will be both players walking away empty handed, which is an inferior outcome compared to the 1 unit gain each could have received by contributing to accomplish the goal.

Although there is no paradox in this result, many people remain skeptical towards PD’s sub-optimal outcome. Some people reason they would play the game differently. They motivate these alternative strategic choices by introducing emotional rewards (“I prefer the other being happy”) or punishments (“if the other free rides, fine: it won’t make him happy”). However, we should not lose sight of the fact that the payoffs in PD are assumed to include all consequences – monetary or otherwise – of the strategic choices made. In other words, one should considering the payoffs to quantify the changes in happiness attributable to the various game outcomes.

However, considering non-monetary gains or losses does translate into a challenge towards PD. One might ask “is a PD-type game even feasible when incorporating real-life non-monetary consequences?”.

I am quite convinced that in human social setting PD-type encounters are the exception rather than the norm. This is because the severity of the consequences resulting from social regulation (retaliation and punishment) are not bounded. If such drivers are present, PD games can be rendered unfeasible. For instance, in a society with strong retaliation morals a PD payoff structure is not achievable.

To see why, let’s consider again the above PD example. We change the strategic choice ‘contribute’ into ‘contribute and retaliate’. Under this strategic choice a player contributes, but if the other turns out to free ride, a fight is started that will cost both parties and that will continue until the free rider’s net gain has dropped to what he would have earned if both had opted for a free ride. Such a change in payoff structure changes the game from PD (prisoner’s dilemma) into the coordination game SH (stag hunt). This change is irreversible: in the presence of retaliative behavior, it is not possible to recover a PD payoff structure no matter how one chooses to tune the monetary payoffs.

Compared to PD, the SH game still carries the same win-win outcome (mutual contribution) with the same return per player, and also still carries the same lose-lose outcome (mutual free ride) with the same zero gain per player. However, in contrast to the PD situation, in SH the win-win situation does form a rational choice (a Nash equilibrium). This is the case due to the fact that, given the choice of the other player, in an SH win-win neither player regrets their own choice. Effectively what happened is that under the win-win scenario the cost that would be incurred by retaliation eliminates any regrets after the fact.

We conclude that PD games cease to be feasible under the threat of retaliation. When including the costs associated with retaliation, what might seem to be a PD game, turns out to be an SH game. By eliminating PD games, retaliation also eliminates the inevitability of suboptimal (mutual free ride) outcomes. The same effect is seen when introducing retaliation options by repeating PD between players. In repeat PD games ‘contribute and retaliate’ strategies (tit-for-tat approaches) dominate over ‘free ride’ strategies.

Retaliation is a disruptive phenomenon. Amongst selfish individuals it eliminates PD type games and helps avoiding them getting stuck in suboptimal outcomes. One might therefore philosophize groups with retaliative behaviors to carry a evolutionary advantage over groups without such a trait. Whether this is correct or not, for sure retaliation is widespread amongst human societies.

In any case: if you feel not at ease with the two-player PD game outcome, your intuition probably is correct. But don’t fall in the trap of translating this discomfort into futile challenges towards the outcome of the game. Instead, challenge the PD scenario itself. I have yet to see a two-person decision scenario that is best describes as PD (rather than as a coordination game).

Now this doesn’t carry over into multi-player games. Many situations of humans failing to cooperate towards a greater good can be modeled as multi-player extensions of PD. But that is a subject deserving its own blog post.