Be A Winner: Ignore The Facts!

Game theory sheds light on the question “how come people get away with ignoring the facts?”

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Facts are our friend when it comes to taking the right decisions. Facts can reveal, ahead of the decision, that a certain choice will inevitably lead to sub-optimal results. Results that will be regretted. Yet we see such ill-fated choices getting pushed ahead on a daily basis. Facts don’t seem to matter anymore.

How do people get away with fact-ignoring decisions? Surely in competitive situations these people will be crushed by those who do consider the facts and take well-informed decisions.

Well… you might think so. But think again. A move guaranteed to be sub-optimal can be a winning move.

Huh?

Let me make this more precise by introducing a simple game. 

Investment Dilemma 
You are part of a group of 500 players who face an investment decision. For each of you there is two choices: invest a dollar or don’t invest. Each dollar invested yields two dollars of return to each and every participant. (Sounds like an amazing yield?Just think of each dollar getting invested in much-needed infrastructure that will benefit each participant.) All of you make your individual decisions at the same time without any opportunity for prior discussion. You are all tasked to maximize your own return.

What do you do? Do you invest one dollar, or will you defect?

The choice should be clear. You can’t influence what the others do, you simply have to live with the return that results from their choices. But you are in control of the return resulting from your own decision. If you don’t invest, it doesn’t cost you, and also it doesn’t contribute to the gain going your way. Your net gain resulting from your decision to defect will be zero. If you do invest, this will cost you one dollar, while your payout gets increased by two dollars. Your net gain resulting from your decision to invest will be one dollar.

That was easy. You decide to invest.

And so do the 499 other players. You will walk out of the game 999 dollars richer. And so does everyone else.

Except… when some players chose to defect. Each player who defects will decrease your return by two dollars. And… each of them will walk out of the game with a net profit that is one dollar higher than yours. In the most extreme case you would be facing the situation where you, and none of the 499 others, invest. You gain one dollar, and all others walk away with double the gain.

Yet it was you who reasoned based on the relevant facts. You chose to invest. Your decision was optimal: you would not have improved your gain had you opted for the alternative. On the contrary, you defecting would have made you leave the game empty handed. All the others, on the other hand, made sub-optimal individual decisions. Any single one of them who would have opted for ‘invest’ rather than ‘defect’ would have increased their own gain by one dollar. And yet, despite their fact-ignoring moves, each of them is walking away with a gain larger than yours.

What is happening is that each player opting for the fact-ignoring choice ‘defect’ is destroying one dollar of their own return and also destroying two dollars of anyone else’s return. Obviously, each and every player who did make the fact-ignoring move ‘defect’ will happily ignore this factual analysis. At best they will walk out of the game with a big smile, at worst they will ridicule you and all others who made the move ‘invest’, and declare all of you to be a bunch of suckers.

Now you might object that this investment dilemma is a rather artificial game deliberately constructed for this effect (ignorant moves scoring better than well-informed moves) to emerge. Surely in a more competitive zero-sum game, where your loss is my gain and your gain is my loss, moves destined to lead to suboptimal result can not be winners!

Hmmm, are you sure? Let’s consider the following multi-player zero-sum game.

Majority Prediction

You are part of a group of four players. Each of you is facing the same two choices: choose “2” or choose “3”. All four of you make their choice simultaneously without any opportunities for communication. You win if you have correctly predicted the number of players ending up under the majority choice. Three sets of outcomes need to be considered:

  • If two players chose “2” and two players chose “3”, the majority size is 2. So the choice “2” wins and two dollars in total move from the players who chose “3” to the players who chose “2”. The losers contribute equally and the money is spread evenly over the winners.
  • If three players choose “2” and one player chooses “3”, or if three players choose “3” and one player chooses “2”, the choice “3” wins. In these cases six dollars move from the losers (those who chose “2”) to the winners (those who chose “3”), again with the losers contributing equally and the winners benefitting equally.
  • If all four players choose “2”, or all four players choose “3”, there is no winner, and no money gets transferred.

The various outcomes with their payouts are summarized in below figure.l


Each of the players is tasked to maximize their individual gains. How would you play? Do you choose “2” or do you choose “3”?

If you consider the facts and inspect the payoffs in above figure, the decision is easy. Regardless which combination of choices is made by the other three players, the choice “3” always leads to a higher gain than the choice “2”. For instance, if two opponents choose “3” and one opponent chooses “2”, you choosing “2” will lead to a gain of $1, while you choosing “3” will lead to a gain of $2. Choosing “3” leads to a better result than choosing “2”. You can check for yourself that this is always the case.

Considering these facts, you choose “3” without the slightest reservation. And you expect all opponents to arrive at the same choice. That would lead to each of you leaving the game empty-handed, an outcome to be expected for a symmetric zero-sum game. If everyone plays optimally no one will win.

Now considers what happens when amongst your three opponents two players ignore the above facts and make the ill-informed choice “2”, while the third opponent chooses “3”. You stick to the well-informed choice “3”. The result is that the fact-ignorers both walk away with a gain of $1, at the expense of you and the other player who made the rational choice. And the thing is: you don’t regret your choice. Had you made the choice “2”, given the choices of the others your loss would have been $2 instead of $1. Also in hindsight you have made the optimal play. On the other hand: each of the players who chose “2” has to live with the fact that they could have won $3 instead of $2.

Of course they don’t care. They won, you lost.

Let’s modify the game somewhat in order to make this effect (fact-ignorers defeating rational players) more pronounced. You are in the same game, but now the choices are not made simultaneously but in turn. The first player chooses “2”, and the second player also chooses “2”. You are in third position and it is your turn. What do you do?

The key observation to make here is that the game hasn’t changed really. The fact that decisions are made in turn is basically a distraction. The bare facts (the rules for the payouts) still dictate choice “3” for all four players participating. This is easy to see for the fourth player, but then it also follows for the third player, and so on.

The first player ignored the above facts when choosing “2”, and so did the second player by also opting for “2”. Now it is your turn. How do you extract maximum benefit from the errors made by the first two players? By now you know the answer: you have to select “3” and so should the fourth player. But let’s spell out the detailed reasoning. If you would choose “2”, the last player will opt for “3” and pocket a gain of $6. You lose $2. If instead you choose “3”, the last player will also select “3” to limit his loss to $1. As a result you also lose $1.

So you and the fourth player, who both carefully consider the relevant facts (including the fact that the first two players did choose “2”) will both choose “3”. As a result you both lose money to the two players who ignored all facts and played sub-optimally. You have found yourself in a zero-sum situation where you can’t benefit from the errors that others have made.

Let’s analyze the situation from start. By selecting “2” the first player makes a blunder. If all subsequent players play optimally by selecting “3”, the first player walks away with a loss of $6, and all other players gain $2. Now by selecting “2” the second player also makes a blunder. Instead of walking away with a gain of $2, he or she will leave the game with a gain of at most $1. Interestingly, while this blunder comes with a self-harm of a $1 drop in gains, the harm to each subsequent player is thrice as large. As a result the first player receives a gift in the form of a gain increase of $7. Such a multiplayer gain transfer is known under the name Morton’s effect, named after Andy Morton who first described this phenomenon in the game of poker.

The conclusion of all of this? Decision situations in which multiple parties interact and compete are tricky business. You can’t assume that parties who consistently disrespect the facts will necessarily accumulate a self-harm that puts them at a disadvantage. Even in simple and clear zero-sum encounters, parties ignoring the facts can thrive over those who carefully weigh all the facts. Ignorance does inflict self-harm, but the collateral harm to rational competitors can far exceed the self-harm. Particularly when the fraction of ignorants reaches a critical mass, this effect can become most pronounced.

George Carlin’s warning “never underestimate the power of stupid people in large groups” holds true, certainly so in this day and age.

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.