Game Theory, Prisoner’s Dilemma, Nash Equilibrium, Stag Hunts, and Sherlock Holmes…and Counter-Terrorism? 1/2

Game Theory

In my previous post, I talked about Game Theory and my views on how it could be applied to eBay auctions.  In this post, I plan to broaden my explanation of the idea and relate it, along with ‘brother theories’ to Counter-Terrorism policies.

What is Game Theory?  Well, to quote the presentation by Professor Donald Shemanski in SRA 211, a class that I am TI-ing, Game Theory presupposes rational actors making strategic decisions, while attempting to anticipate the response of their opponents.  Players try to outguess their opponents by imagining how they themselves would act if they were in their opponents’ position.

This theory can be difficult to understand at first, but once you look at it long enough, the thought process becomes clear.  In class, we played a game where one pair of people decided whether or not to throw an X or a Y.  In their team of 4 pairs, each pair did the same and compared guesses after each round.  Points were awarded in a way that tried to force pairs to always throw a Y.  However, if only one team threw the X, they were rewarded and the other pairs lost.  After many rounds, about half of the class ended up in the positive.  The idea behind this exercise was to show that, if you worked together, your whole team would win, but if your pair decided to screw the team, the pair would win while the team lost.  It all comes back to the idea that one pair could not win on their own, the whole team had to agree and participate to win.

This theory was also explained on pages 238-9 in the book Against the Gods: The Remarkable Story of Risk, by Peter Bernstein, a book I am reading for my SRA 311 class with Will McGill.  Image this scenario, you are offered a guaranteed $1, or a 50/50 chance of winning $2 or nothing.  The expected utility here is $1 ((.5 x 2) + (.5 x 0)).  You cannot expect to make more than $1, on average, so you should take the dollar and go.  Now imagine you are offered $100 guaranteed, or a 67% chance of winning $200, and a 33% chance of winning nothing.  Your expected utility is $133 now, so you should take the gamble because you can expect to make more than the guarantee on average.

Last year, in my Micro-Economics class, we did a similar exercise that is also explained in my previous post.

Still confused as to how this pertains to Counter-Terrorism?  Continue to read as the following theories lend themselves to this one and it will become more clear.

Prisoner’s Dilemma

This Dilemma was first introduced to me in my Micro-Economics course by Dirk Mateer, then in my International Politics course via Errol Henderson.  Now that Don Shemanski has lectured on it, I figure it is a very important topic.  Like Game Theory, it is simple to understand, even if it does seem so at first.  For this blog post, I will explain it the way Henderson taught it in Political Science 014.  While most examples use Prisoners for the example, my explanation will talk about states in an arms race.

State one, we will use Pakistan, is mulling over a decision to defect or accept an anti-poliferation policy passed down by the United Nations.  State two, India, is faced with the same decision.  These two states have been at odds against each other for quite some time now, so the strategy of one directly effects the strategy of the other.  We learned that the states would use a strategy called Tit-for-Tat.

What does that mean?  Well, look at the chart below.  If they both accept the policy, both states will get a 5 and be the best off, both without weapons.  However, if India accepts first and Pakistan is yet to make a decision, what choice should they make?  Well, they can either accept, giving India a 5, and themselves a 5 (Red), or they can defect, giving themselves a 10 and India a -5 (Green).  The choice is obvious that they would defect and get ahead, leaving India worse off than at the beginning (India would have no weapons and Pakistan would have all of theirs).  Because neither state wants to make the first choice, because the opposite state will most likely screw them over, both states would defect, keeping all their weapons (both getting a 1) and being no better off than before.

		India
		accept	defect
	accept	5(P),5(I)	-5(P),10(I)
Pakistan	defect	10(P),-5(I)	1(P),1(I)

The example used in Shemanski’s and Mateer’s classes were about prisoners, but hold the same concept.  If you have two prisoners and they are given the option to confess or not confess to a crime, what would they do if they know that confessing will give them freedom?  They should both confess and even though they will never ultimately get their hands on the loot, neither will end up in jail.  But if prisoner one does not confess and prisoner two does, prisoner one gets all the blame and rots in prison.  If both do not confess, they are the best off, but who would take the risk that the other guy would also not confess?   A great question was asked in class, what if the confession of prisoner one leads the prosecution to believe that prisoner two had no part in the crime.  Dirk Mateer would scream that this is in violation of Ceteris Paribus!  He stressed this from day one, it is when you bring an external factor into a question that you shouldn’t.  Essentially, it is overthinking the question at hand, but a good question none-the-less.

		Prisoner Two	Prisoner Two
		not confess	confess
Prisoner One	not confess	5(P1),5(P2)	-5(P1),10(P2)
Prisoner One	confess	10(P1),-5(P2)	1(P1),1(P2)

Nash Equilibrium

If there is a set of strategies with the property that no player can benefit by changing her strategy while the other players keep their strategies unchanged, then that set of strategies and the corresponding payoffs constitute the Nash Equilibrium.

Look below at the table denoted “Stag Hunts.”  This theory, developed by Rousseau from Switzerland, shows a Nash Equilibrium.  In this theory, both hunters are needed to hunt the stag, if one chases the hare, the other cannot kill a stag.  Because the optimal outcome relies on both participants, a nash equilibrium is created.  If both hunters decide to chase the hare (Red), then hunter one has no incentive to hunt the stag instead.  He would be moving down from 1 to 0 (Green).

Stag Hunts

		Hunter Two	Hunter Two
		hunt stag	chase hare
Hunter One	hunt stag	3(H1),3(H2)	0(H1),2(H2)
Hunter One	chase hare	2(H1),0(H2)	1(H1),1(H2)

Ultimate Question

Finally we come to the ultimate question.  In this table, no nash equilibrium is created because there is always incentive to change your decision based on the decision of the oppositioin.  Should Moriarty get off his train at Canterbury, and Sherlock Holmes also gets off at Canterbury, Moriarty will kill him (Red).  But, if Holmes get off at Dover, and Moriarty gets off at Canterbury, Holmes runs free and lives happily (Green).  This is a classic example of a zero-sum problem.

		Moriarty	Moriarty
		Canterbury	Dover
Holmes	Canterbury	-1(H),1(M)	1(H),-1(M)
Holmes	Dover	1(H),-1(M)	-1(H),1(M)

Now you may be asking yourself, why should I care about these theories and how do they connect themselves to Counter-Terrorism?  Well, the point behind this, part one of two posts, was to introduce the topics.  Part two will focus on their application to Counter-Terrorism.  If you have any questions, or comments, please comment below and get the discussion started!

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Possibly related posts: (automatically generated)

• Todd Sandler on Game Theory, Terrorists, and International Relations 2/3
• “The Trap” or Game Theory from the Nash Equilibrium predicts the future