Terrorism and Game Theory

The interactions between terrorist and their "targets" make game theory perfect for understanding terrorist behaviors. Terrorism is the use of violence and intimidation in the pursuit of political aims (google). Since terrorist are active players these models below do not cover all complexities, but the more things these scenarios take into account the more accurate they become. Game theory is used to gain insight into terrorist behavior and hopefully lead to helpful counterterrorism plans. Examples of game theory in terrorism are seen below:

Game theory in the idea of deterrence cost:
Scenario 1:
1) home country
2) foreign country
3) terrorist group
Goal: to prevent terrorist attack on your country
If the home country increases their deterrence efforts, the cost of a terrorist attack on the home country will also increase, therefore the relative cost of attacking the foreign country will be decrease, leaving the foreign country more vulnerable, in which case they need to increase their deterrence efforts. This will again even out the cost of terrorist attack for both countries so the home country will again increase there deterrence efforts, and this becomes an endless cycle and leads to overspending on deterrence efforts.

In this example if the two countries were non cooperative they would choose to differ, in reality, the optimal choice is to cooperate and preempt






Scenario 2:
1) 2 allied countries
2) terrorist group
Goal: allied countries to freeze terrorist assets
desired outcome can only be achieved when both countries choose to freeze assets so they must be cooperative








Scenario 3:
1) home country
2) foreign country
3) terrorist group
Goal: intelligence gathering and sharing  without discovery

no collaboration vs collaboration

Game theory and the idea of Terrorist Attacks:
Scenario 1:
2 distinct terrorist subgroups that are part of the same overall group, no chance of failed attack

terrorist net pay out for an attack: +4
+4 (pride points)  +1 (resource point from supporting nations) -1 (cost of attack) = 4

terrorist net payout for an attack done by other terrorist group: + 1
+2 (pride points) - 1 (resource points) = 1

dominate strategy: both groups to attack


Scenario 2:
takes into account short term change in military allocations based on actions of the terrorist groups

m_A - percent of military allocations used to suppress terrorist A

m_B-  percent of military allocations used to suppress terrorist B

note: m_A + m_B = 1, terrorist receiving 100% of military attention have payout of -1, terrorist receiving 0% have payout of 0

dominate strategy: do not attack

Scenario 3:
Takes into account long-term military allocations, including reinforcements or withdrawals, and strength of military

M - strength of military (0 <= M <= 1)
m_A - percent of military allocations used to suppress terrorist A
m_B -  percent of military allocations used to suppress terrorist B
change in M - attack causes increase in M
- change in M - no attack causes decrease in M


dominate strategy: do not attack

Scenario 4:
Takes into account long term military and possibility of failure in attack


Successful attack payout: (same as scenario 1) terrorist net pay out for an attack: +4
+4 (pride points)  +1 (resource point from supporting nations) -1 (cost of attack) = 4

terrorist net payout for an attack done by other terrorist group: + 1
+2 (pride points) - 1 (resource points) = 1

Unsuccessful attack:
net payout for attack group: -6
-4(shame points) -1(resource points) -1 (resources spent)

net payout for other group: -1
-2(shame points) + 1(resource point)

These different scenarios lead us to a generalized game for two terrorist groups:

which we will decipher tomorrow in class!

One of the four pillars of United States terrorism policy is "no concessions to terrorist". Although this is always the goal, we see from the complexities of just these simple scenarios, that this is not always as easy as it sounds.

The Greek Debt Crisis

The Greek Debt Crisis is an event that has had an impact on the European and Global Economy throughout the past decade. I have only really paid attention to this issue in the past year after taking courses in Global Economics and studying in Europe, where the Greek Financial Crisis is an ongoing issue. Most of you have probably heard something about Greece this past summer as a lot of decisions were made about whether the European Union would continue to bailout Greece and loan them money to make up for the massive debt it has accumulate over the past 15 to 20 years. For those of you who don’t know much about this issue at all, here is a brief overview of the situation. Basically, it all began when the European Union created a Eurozone that was put into place in 2001. The Eurozone is a monetary union that consists of 19 countries in the European Union that all agreed to adopt a common currency, the Euro. Greece is one of these countries. However, in order to become eligible for the Eurozone, a country must meet specific financial requirements. Apparently, Greece already had a fair amount of debt before the creation of the Eurozone, but the Greeks had lied about their government budget deficit at the time in order to be accepted into the Eurozone. This wasn’t a problem for the European economy at first, but the global market crash of 2007-08 exposed all of Greece’s financial problems due to a slowing economy that affected the tourism in Greece which Greece’s economy was very reliant on. Many different bailout packages and failure to meet repayment dates to pay off its debts has led to a huge debt crisis that is having an impact on the entire Eurozone. 

  

 Prisoner's Dilemma in the Greek Debt Crisis

So how does this relate to game theory? There are a number of ways which game theory can be applied to the Greek Debt Crisis. Recently, many economists have been trying to relate the Prisoner’s Dilemma to the current situation in Greece. The players in this game are Greece and the European Union. Both Greece and EU must make decisions about the financial crisis in order to determine what will happen to the future of the European economy. The outcome is dependent on both Greece’s and the EU’s strategies. Greece developed a bailout plan that would incorporate new taxes on the wealthy in order to reduce its spending cuts while also trying to pay off its debts. Greece would still need the economic assistance and approval from the EU in order for this plan to be put into place. 

The strategies for Greece are to either:

1)      Accept its plan and accept the continued assistance of the EU

2)      Be forced to leave the Eurozone, because Greece defaults

The strategies for the EU are:

1)      Accept Greece’s bailout plan and continue to bailout Greece

2)      Reject Greece’s bailout plan and force Greece out of the Eurozone

The decision tree below explains the three possible outcomes for the above strategies.

1) As you can see, the most beneficial option for Greece is to offer its bailout plan and ask for assistance from the EU. The EU would then have to accept this plan in order for Greece to receive maximum payoffs. The EU's payoffs would not be as high as for Greece in this situation, because the EU would continue to bear some of the Greek debt burden. The payoff for this outcome is (1, 3/4).

2) The most beneficial outcome for the EU would be to reject Greece's bailout plan with the assumption that Greece would solve its debt problems internally by leaving the Eurozone, but uncertainty and outside factors cannot ensure this outcome for the EU. The payoff for this outcome is (0, 1).

3) If the EU rejects Greece's plan and Greece leaves the Eurozone, this could be detrimental to the European economy by causing the Eurozone to collapse. This is the worst possible outcome for both Greece and Europe. The payoff for this outcome is (0, 0)

Because the existence of outside factors have a large impact on whether a Grexit would impact the entire European economy, there is no way to avoid a Eurozone collapse if Greece were to leave the Eurozone. This is why the EU would choose to accept Greece's plan every time in order to avoid the worst possible outcome for both the EU and Greece. The Nash Equilibrium in this case would be if Greece offers its plan and the EU accepts it with the payoff of (1, 3/4). 

This example differs from the Prisoner's dilemma in a few ways. First, this is a cooperative game as opposed to the a non-cooperative game in the Prisoner's Dilemma. Greece and EU are expected to cooperate and discuss their strategies and possible outcomes. Also, the Prisoner's Dilemma is a symmetric game which is a game where one player's payoffs can be expressed as a transpose of the other player's payoffs. This is not the case for the Greek Debt Crisis, so we call this game an asymmetric game. 

Coming soon...

There are many more ways to apply game theory to this complicated financial crisis that is having a large impact on the world today. Lucky for all of you, you will be hearing about some more applications of game theory to the Greek Debt Crisis in just a few weeks. I am writing my final paper on this topic, so hopefully you can all learn a little bit more about this application. Let me know if you have any questions or concerns so far, and I'll try my best to answer them!

Sources:

http://www.bbc.com/news/magazine-33254857

https://www.stratfor.com/analysis/john-nashs-legacy-mathematic-theory-strategic-implications

http://www.nytimes.com/2015/06/05/business/in-greek-debt-puzzle-the-game-theorists-have-it.html?_r=0

http://www.nytimes.com/interactive/2015/business/international/greece-debt-crisis-euro.html

http://www.investopedia.com/articles/investing/071415/game-theory-and-greece-bank-crisis.asp

Game Theory in The Dark Knight!

I personally love when I find places that Math relates to everyday life. And while The Dark Knight might be an extreme relation to real life, it is still a movie, so why not talk about it!

At this point, I hope that everyone who has an interest in seeing the movie already has, I don’t want to ruin the plot for anyone. So if you really want to see the movie, I guess I am assigning you homework to watch it tonight, or at least read about it. The Link below to the blog has a good overview of the movie before getting into the Game Theory discussion.

Game Theory of the Dark Knight 

So without further delay, here is the game theory in The Dark Knight, or at least some of it (there is a lot).

For those who have seen it, I am sure that the first thing that you think about is when the Joker fills the two ferries with people. One ferry carries prisoners from Gotham while the other carries innocent civilians. But here is the twist that gives us the game theory! The Joker has rigged each of the ferries with explosives; he then precedes to hand each of the ferries the detonator to the other ferry. If neither ferry blows up the other ferry before midnight, than the Joker will blow up both of the ferries and everyone will die.

So lets take a look at the payoff matrix for this situation:

So we see that the highest payoff is not dying, for both the prisoners and the civilians.

The civilians recognize this and take a vote on whether or not to blow up the ferry carrying the prisoners. When the vote is tallied, the result is to detonate the ferry. Seems simple in practice, press a button, kill prisoners that have “had their chance,” and you get to live. But it is not as easy as that. There are other factors that go into play. We are going to try and understand these factors by taking a look at a different type of hostage dilemma.

Hostage Dilemma

Lets paint the scene:

-50 people locked in a room by a bad guy

-Bad guy needs a password from at least one of them

-Bad guy will ask hostages one by one what the password is

            -Hostage spills the information, “game” ends

            -Hostage stays quiet, he dies

Now for the payoffs:

Sacrificial and Selfish types

Sacrificial receives -1 from dying and -9 for giving away the password

Selfish receives -1 from dying and does not care if the password is given up

For this case we will make the hostage sacrificial 95% of the time, and selfish the other 5%.

So how often will the bad guy get the password? Turns out that it is 100% of the time, because the first person will always give up the answer.

 Proof

How:

Case 1, the first hostage is selfish.

They will receive a -1 payoff if they die, and a 0 payoff if they give up the password. And thus they will always talk, and so that wraps up case 1.

Case 2, the first hostage is sacrificial.

No one talks, then they die and receive a -1 payoff, if they talk they get a -9 payoff.

Seems simple, but you need to take a look at the future, aka the people to go afterwards. Selfish type will always talk, and so the best-case scenario is that all sacrificial types remain silent. The probability that no one will talk is .95^49=.08.

And thus the probability that someone talks 1-.08=.92, or 92%.

So now this allows us to take a look at the expected utility.

EU(keep quiet)=(-1)(.08)+(-1+ -9)(.92)=-9.28  

EU(talk)=-9

And so the hostage will receive more of a payoff if he/she decides to talk, even though it is –9. And sadly the bad guy will always get the password.

So in the Dark Knight, the hostages, the civilians and the prisoners, have two decisions, detonate or be detonated.

So while the connection between the two is not exactly clear and straightforward, it still helps us to understand that there is more behind the decision.

The civilians who are selfish want to detonate, and the sacrificial ones will not detonate. Detonating will not receive any negative payoff, but sacrificial ones will, and also sacrifice the rest of the ferry.

So I hope this gives you an overview of the hostage dilemma and an intro to the Dark Knight. I hope I didn’t spoil anything yet, but warning there will be a spoiler in class tomorrow.

Thanks guys,

Please post any questions that you have, and I will do my best to answer them in the comments, or in the presentation tomorrow.

Sam Horan

The St. Petersburg Paradox

Introduction

The St. Petersburg Paradox, also known as the St. Petersburg Lottery, appeared in the Commentaries of the Imperial Academy of Science of Saint Petersburg in 1738. It was presented and resolved by Daniel Bernoulli who resided in the eponymous city at the time. Nonetheless, the problem was first introduced by Daniel’s brother Nicolaus Bernoulli in a letter to a French mathematician in 1713.

The Bernoulli Family

The Bernoulli family generated many prominent mathematicians and Daniel and Nicolaus were one of the most prominent. They were Swiss although Daniel was actually born in the Dutch Republic. Nicolas was the older brother born in 1695, but he died early in 1726. He actually taught mathematics to Daniel who was born in 1700 and lived a long life dying in 1782. Daniel was most known for his work in fluid mechanics and in probability and statistics. Leonhard Euler was actually a student of their father.

The Paradox

The problem arises from a game of chance where a coin is tossed with equal ½ odds of yielding heads or tails. The payout starts at 2 dollars and every time a head appears, the pot is doubled.

The expected value of the game is thus the following:

EV = ½ * 2 + ¼ * 4 + 1/8 * 8 + 1/16 * 16 + …

      = 1 + 1 + 1 + 1 + …

      = ∞

Assuming the “casino” where the game is played has unlimited resources, the expected payoff for playing the game is infinite. Due to this unlimited theoretical payoff, one should be willing to pay any sum of money to play the game. However, research showed that most people were not willing to pay even small sums to play the game. The paradox is the discrepancy between the infinite payoff and the people’s willingness to pay to play.

Bernoulli’s Take

His resolution had to do with the difference between expected value and expected utility. He introduced a utility function, the concept of expected utility (which was referred to moral expectation versus mathematical expectation back then) and the concept of diminishing marginal utility of money.

The utility function is the function that take into account people’s preference. What Bernoulli put forward with his expected utility theory is there is something other than expected value which people look at in situations with uncertain outcomes. This something is the impact of the outcome of the gamble on the person taking the gamble, or the impact on his happiness. Finally, the concept of diminishing marginal utility of money means that the more money you have, the least impact an extra dollar will have on you. If we think about it, this is very intuitive. Let’s look at two situation:

11)      Suppose that it’s the first week of the semester and you had a sick job this summer which compensated you handsomely and you have $3000 dollars saved for the semester. Then someone offers you to fill in for his shift as a line judge at the football game. You’ll get paid $30 dollars.

22)      Suppose that it’s the first week of December, you are back from Thanksgiving break. You have had a blast of a semester so far, but you had expensive taste and you have only $150 dollars left from your summer. Then someone offers you to fill in for his shift as ball boy at a tennis tournament in the fieldhouse. You will work for 3 hours on a Sunday morning and will make $25 dollars.

It is very likely that you would not fill in for your friend in situation 1, but probably would in situation 2, and that is despite the fact that the payoff in situation 2 is less than in situation 1. However, that $25 has a much bigger impact on your holdings when you have $150 than $30 when you own $3000.

Here we can see a graph of a relationship where utility is marginally decreasing as X increases.

Source: https://apecon3.wikispaces.com/Diminishing+Marginal+Utility

Bernoulli himself put forward a common utility model which is the logarithmic function U(w) = ln(w) where U is utility and w represents a gambler’s wealth. His utility function attempted to determine the cost someone would be willing to pay to actually play the game aforementioned. The expected utility of the game, given a cost to play of c would be the following:

Following this formula, we learn that someone with a million dollars would only be willing to pay up to $10.94 to play the game and someone with a wealth of $2 would be willing to pay up to $2.

Conclusion

Bernoulli’s take on the paradox was studied and many mathematicians improved his work. However, he pioneered utility theory and its application in mathematical modeling of behaviors of people in situations where the outcome is uncertain.

Looking forward to our discussion in class tomorrow!

Cheers,

Sam

Works Cited:

https://www.math.hmc.edu/funfacts/ffiles/30001.6.shtml

http://mathworld.wolfram.com/SaintPetersburgParadox.html

http://www.policonomics.com/saint-petersburg-paradox/

https://www.jstor.org/stable/2722712?seq=1#page_scan_tab_contents

https://www.jstor.org/stable/1909829?seq=1#page_scan_tab_contents

https://mises.org/library/daniel-bernoulli-and-founding-mathematical-economics

http://www-history.mcs.st-andrews.ac.uk/Biographies/Bernoulli_Nicolaus(II).html

https://en.wikipedia.org/wiki/St._Petersburg_paradox#cite_note-2

The Prisoner's Dilemma

Story Time!

Two members of a notorious gang are arrested and imprisoned. Upon capture, they are immediately placed in confinement with no means of communication with one another. The authorities are having a hard time gathering evidence to convict and accuse these criminals of their main charges. They would like to have both of them sentenced to a year in prison on a lesser charge, but at the same time, the authorities offer each prisoner to a possible deal and bargain. Each can either 1) Betray the other by testifying that the other committed the crime, or 2) to cooperate with the other by remaining silent. The offer stands as follows: 

1. If A and B both betray each, they will both serve 2 years in prison. 
2. If A betrays B but B remains silent, A will be set free and B will serve 3 years in prison (and vise versa)
3. If A and B both remain silent, both of them will serve 1 year in prison only for the lesser charge

Where clearly 0 > 1 > 2 > 3 .... right? 

One thing that makes this game so interesting is the Prisoners inability to communicate with each other. If they were able to, then you would assume that they would both remain silent. With each prisoner being allowed to remain silent, the two could both decide to do so and earn the lowest possible jail time for the both of them.

** But where is the fun in that? This game would be wayyyyy too boring if they could talk. **

Oh Shaggy, just take responsibility sometimes! 

So because they are not allowed to communicate, lets start with some scenarios!

From the perspective of Prisoner A, Prisoner B can either cooperate or defect. If:

1. Prisoner B has decided to cooperate. Then Prisoner A has the choice of also cooperating and giving him a one year sentence, or to defect, setting him free. Logically, Prisoner A should choose to defect, as it means he spends the lesser time of the two options in prison. 
2. Prisoner B has decided to betray. Then Prisoner A has the choice of once again cooperating and thus serve three years, or to also defect, giving him 2 years of jail time. Looking at this logically, Prisoner A chooses to defect, which would give him less time in jail. 

Prisoner B faces the exact same situation in both scenarios and would rationally come to the same conclusion, and this is how the Nash Equilibrium (which is the theory that states that ones strategy isn't influenced by knowing what the other player will choose) is used in this dilemma.

After using the Nash Equilibrium, we come to the conclusion that the best route for each Prisoner is to actually defect, which would get them both 2 years in prison. The interesting thing is that although this choice was the most rational option, there is a better possible payoff if they both decided to cooperate. 

Lets Play Again! Lets Play Again!

In most dilemmas, there are many assumptions that seem to take place, which is actually the main reason why the less desirable choice ends up being the most "rational" choice in the Nash Equilibrium. But lets playing this one more time....

The Iterated Prisoners Dilemma  is when the game is played more than once in succession, while of course, remembering the previous moves of the other player. Because they remember the previous moves, they can change their own choices based on that info. 

For this version, we say that 2R > T + S where R is cooperation, and T and S are the outcomes when each prisoner doesn't choose the same option. So, a mutual decision of cooperation always yields a better outcome than an alternation (one choosing defect and one choosing betray). 

Lets say this game is played X amount of times.  Once again, cooperating would be the most logical for both players to choose and its the most fair, and it would allow a binding trust between the two players to decide to cooperate for each game. However, defecting gives that player the best possible option if you have "earned the other players trust" and want to betray him, but will also motivate to not make anymore future "nice" (cooperating) moves. Sadly, there is no strategy to how these multiple games should play out, but analyzing the game from the last "turn/game", we see that the iterated dilemma doesn't differ from the original. 

So consider the very last turn of X. There are no more turns after this round, so you (in theory) want to look out for yourself because the worry of losing the other players trust isn't a factor anymore. Being the evil person we all naturally are, we would choose to defect as there is no retaliation later, and would earn the highest payout reward. Of course, monkey see, monkey do, as this process is copied by the other prisoner. Using this same logic, it would also make the most sense to defect on the second to last turn as well, since the last turn is already set in terms of what move you will do. Trickling this down all the way down to the first turn, where you will once again defect, the strategy is the same for both the iterated and regular versions of the Prisoners Dilemma.

Lets Lock it up! (Get it?!? Does that play-on-words work?Let me know...)

The Prisoners Dilemma is a great practical use of Game Theory and with the use of the Nash Equilibrium, we get to see the strategy that would yield the "safest" outcome for both players. There have been many variations to this game which allow for the players to grab a new twist to the concept, but all of the theories and applications of Game Theory and Nash Equilibrium remain the same. 

Hope you enjoyed!

Jon 

References: 

https://cdn.andertoons.com/img/toons/cartoon4053.png

https://en.wikipedia.org/wiki/Prisoner%27s_dilemma

https://31.media.tumblr.com/90fe57665c665cf9aebbb1f34aeae151/tumblr_inline_n04negmVba1qzo1my.gif