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.

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