The All-Seeing Eye

Musings from the central tower…

Traveler’s Dilemma and Opportunity Cost

As a followup to my last post, I thought now would be a good time to say some things about opportunity cost, or The OC.  Please do not confuse this with any other things called The OC.

Opportunity Cost is an economic analytical tool – a measure of the cost of a missed opportunity.  It goes like this.  Let’s say you have a dollar and you’re standing on the street at a hot dog cart.  The cart is selling pretzels for $1 and hot dogs for $1.  If you buy the hot dog, you can’t buy the pretzel.  Therefore the opportunity cost of buying the hot dog is one pretzel.  Conversely if you buy the pretzel you can’t buy the hot dog.  So the opportunity cost of the pretzel is one hot dog.  Simple, right?  At first glance this seems a trivial and reductive measure for an economist to be thinking about, but in real life, when applied to more complex situations, we can see the value of considering the opportunity cost.

Let’s try another example.  You have a dollar but you aren’t hungry, and you’ve got a year.  You can keep the dollar in your pocket and at the end of the year you’ll have a dollar.  Or you can put the dollar in a bank and at the end of the year you’ll have, let’s say, $1.05.  The average person would think that if they kept the dollar in their pocket, they haven’t lost anything, and in one sense this is true.  However, they have missed something – the opportunity to earn $.05.  The opportunity cost of holding onto the dollar was five cents.  Doesn’t seem like much, but what if it’s a thousand dollars?  A million?

The point is, when there’s money at stake it pays to consider the opportunities that you have when making choices, because in some sense, missing the opportunity to earn money is sort of like losing money, even if it’s money you never actually had.  An opportunity is worth something.  If you don’t believe me, play poker.  If you fold a hand, and it turns out at the end that you would have won if you had stayed in, you will feel like you have lost something.  What you’ve done is missed an opportunity, and the loss you’re feeling is the opportunity cost.  Folding may have been the right decision based on the odds, but you’ll still feel bad that you didn’t get the pot.

So what does opportunity cost have to do with the Traveler’s Dilemma?  Well, it’s another way of evaluating possible plays in the TD, and it demonstrates a major flaw in the models used by game theorists to “solve” the TD.

To recap, in the TD, game theory says that logically speaking, a player ought to play (2).  Many people intuitively feel that they should play higher, and (100) is perhaps the most common play, with (95) to (100) comprising the majority of plays in some experiments.  According to Kaushik Basu, (2) is the correct or best play, because of a game theory concept known as the Nash Equilibrium.  Further, (100) is the worst play, because it is the only play in the game that is “beaten” by every other play.  In other words, if player 1 plays (100) and player 2 plays anything else, player 2 will be rewarded more points than player 1.  If this logic is to be believed, (2) is a better play than (100), and people, if they are acting “rationally,” ought to play it.

But let’s look at the OC to see if that’s true.  Let’s say player 1 plays (100) and player 2 plays (2).  The rewards, then, are 0 points to player 1 and 4 points to player 2.  However, given player 2’s play of (2), the highest score player 1 could possibly have acheived by making a different play is 2, by playing (2).  Any play other than (2) results in a score of 0.  So, player 1 lost 2 points by not playing (2), or, to put it another way, his opportunity cost for playing (100) was 2 points.

Player 2, however, is in a much worse situation.  He played (2) and got 4 points.  Given player 1’s play of (100), the highest score possible for player 2 was 101, with a play of (99).  In other words, by playing differently player 2 could have gotten 101 points, but instead he got four.  That means that his opportunity cost for playing (2) was 97 points.

So, in the above example, on the face of it it seems like player 2 won – he got 4 points, while player 1 got none.  However, if you look at it a different way, player 2 lost 97 points while player 1 only lost 2.  If you consider the scale of a loss of 2 vs. a loss of 97, you see that a play of (100) is much less risky than a play of (2).

In fact, for an opponent’s play of 2, the OC of (100) is 2.  For any number between (3) and (99), the OC is 3.  And for (100), the OC of (100) is 1.  The OC of (2), however, is (Opponent’s play – 3).  That means that for any play above (6), the opportunity cost of (2) is higher than that of the opponent’s play.

So the (2) player almost always loses more money than his opponent – not that the players are losing money that they actually possessed, but money that they could have – and perhaps should have – earned.  If you ask any economist or poker player, that loss can sting just as much as a loss of cold hard cash.

The thing is, if you evaluate the Traveler’s Dilemma in terms of Opportunity Cost, the definition of improving one’s position changes, and therefore so does the Nash equilibirum.  It’s a situation where gaining money and not losing opportunity are not the same thing – and this situation probably comes up fairly often in the real economy, which is why opportunity cost is important as an economic concept.  Rational choices and selfishness, therefore, cannot necessarily be evaluated successfully using only the rubric of amassing the most gain by the end of the game.  There are other measures of success, and people do use them.  Game theorists and economists alike would do well to remember that.


January 25, 2008 - Posted by | Economics, Game Theory | , , , ,

1 Comment »

  1. […] When looking at an exchange, what is really being exchanged?  Let’s use the Traveler’s Dilemma as an example.  The game theorist expects a “rational” player to play (2), assuming […]

    Pingback by Economics Foundation « The All-Seeing Eye | March 8, 2008 | Reply

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