Game Theory Dilemma: Too Much Is Not Enough

We’d like to offer our bittersweet congratulations to the late Heath Ledger for earning an Oscar nomination for his role as the Joker in Batman: The Dark Knight. As fans of the Cape Crusader well know, Ledger’s character is responsible for establishing the central plot twist of the film, a dilemma shared by ferry boats of ordinary civilians and incarcerated prisoners over whether to save themselves by cruelly slaughtering others.

Economists around the world have noted with some surprise, and a fair amount of pride, that this plot device accurately portrays a classic scenario of game theory known as the Prisoner’s Dilemma. In its original format, the scenario features a pair of prisoners who are each being pressured by police investigators to “squeal” on the other, and who must each decide whether they can trust the other to maintain silent fidelity.

Like many game theory scenarios, the Prisoner’s Dilemma focuses on choices that flawed human beings must make about whether to cooperate or to compete … to be faithful or to mistrust … to share with others or to grab for themselves. Interestingly, many seemingly incomprehensible macroeconomic conditions can often be understood by referring to these simple models of human behavior.

Rolling The Dice

Take the boom-and-bust cycle of the Las Vegas hotel market, for instance. Surely the developers of hotels and casinos must realize that there is a limit to their market’s demand for slot machines and table games! But if they realize this simple truth, then why are so many mega-projects still moving inexorably towards completion at a time of deep recession?

Wynn Resorts grabbed a head start on its rivals by opening its $2.3 billion Encore in December 2008; fast in pursuit for openings during 2009 and 2010 are the $9.1 billion CityCenter by MGM Mirage, the $3.9 billion Cosmopolitan, the $2.9 billion Fontainebleau, and a significant room expansion by the Hard Rock Hotel. One analyst noted grimly that it’s going to take years to digest the … capacity; in the meantime, room rates may drop well below the $100 level …

… hello, $25 a night stays!.

Urban planners may wonder how business developers could have signed off on all of these projects while failing to anticipate how their simultaneous expansions of room inventories would depress retail room rates. Not every one is falling into the same trap; because of such concerns, Caesars Palace by Harrah’s has partially cancelled its growth plans. Nevertheless, most of the major firms are blindly sailing ahead into the headwinds of a severe economic bust.

How Game Theory Works

Game theorists have a brief explanation for this example of mass myopia; they simply believe that business developers are fallible human beings. And fallible human beings, they explain, tend to look at the world from their own perspectives; in other words, they usually fail to remind themselves to look at situations from the perspectives of others before making decisions.

The key insight is that people should never make decisions about how to deal with their circumstances until they first consider how others are likely to respond as well. To phrase it a different way, game theorists believe that human beings are self-absorbed and self-focused individuals who aren’t very good at placing themselves in the shoes of others.

The Nobel prize winning economist John Nash developed a model known as the Nash Equilibrium; it can be helpful in illustrating this tendency. The model has been customized by countless thousands of business people and academics during the past few decades; here is yet another customized example..

Let’s assume that you are playing a gambling game in a casino with another player. You and your counterpart must each simultaneously raise one of your two hands. As soon as you do so, you jointly observe which hands are raised in the air – i.e. either (a) your left hand and his left hand, or (b) your left hand and his right hand, or (c) your right hand and his left hand, or (d) your right hand and his right hand – and then you use the following chart to identify the winners and losers.

At first glance, most people would recommend that you should begin by raising your left hand. After all, if you decide to raise your left hand, you are guaranteed to win $1. But if you decide to raise your right hand, you might either lose $2 or win $1. So why risk losing $2, with no prospect of winning more than $1, when you can lock in a guaranteed $1 gain?

In addition, considering that you have no idea what the other player might do, you might decide to assign him a 50% chance of raising his left hand and a 50% chance of raising his right hand. That means that, if you decide to raise your right hand, your overall “expected value” of a 50% chance of losing $2 and a 50% chance of winning $1 is actually a negative value. Thus, raising your left hand and winning the guaranteed $1 remains the better bet.

But what happens if your winning cash receipt is increased from $1 to $1.01 if both parties raise their right hands? Most people would still recommend that you should raise your left hand. After all, the potential loss of $2 remains far greater in absolute terms than the potential gain of $1.01. Why change your recommendation over a potential increase of a measly penny?

Sounds reasonable? I hope not! If you find yourself agreeing with this stream of logic, you have regrettably fallen into the mind trap of self-absorption.

The Mind Trap

To spot the trap, stop for a moment and look at the chart of winner and losers from the other player’s point of view. That’s right; take a minute and focus on his columns and not on your rows. Do you see why this stream of logic is wrong?

The insight is pretty simple. The other player should see that, if he raises his left hand, he is guaranteed to break even. If he raises his right hand, he is guaranteed to win $1. So why wouldn’t he decide to raise his right hand?

Of course, that’s exactly what he will always do. And once you notice that, you will undoubtedly realize that you are always better off raising your right hand – and not your left one – to lock in a guaranteed $1.01 win. Assuming that your winning cash receipt is $1.01 if both parties raise their right hands, of course.

Does this imply that the Las Vegas hotel and casino developers should have avoided making expansion plans in the Nevada market a few years ago, even though the market itself was strong and growing at the time? In a word … yes. Had they each taken into consideration the oversized expansion plans of their fellow hotel and casino developers, they would have undoubtedly realized that they were all building into a market that couldn’t possibly grow forever.

Joining a boom during a cycle of prosperity always seems like a good idea while the boom lasts, but the inevitable bust always follows. In retrospect, the better bet – in other words, the more prudent choice – is usually to invest in the growth opportunities that have not already been discovered by the rest of humanity.