Joker’s game: An extended game of hope and paranoia

I am sure that there have already been a number of game theorists who have commented on Joker’s ship game in the Dark Knight. For example, the following link analyzes the game while factoring in human desires to survive and to obey morality. Also, it introduces a probability that batman will intervene and save the ship.

However, one shortcoming of this model is that the ship game is treated as a one-shot game, which does not take into account the changes that may occur in human thought and behavior over time. For example, suppose that Joker started the game at 6pm, and the ships are set to detonate at midnight. You are aboard one ship, and you know that the other ship has no intention to blow you up. However, you feel morally obligated to keep the other ship alive as long as possible, in case batman does intervene successfully. Would you end up blowing the other ship up? Well, if you ultimately want to live, you would probably want to blow the other ship up a few seconds before midnight, granted that batman has not yet succeeded. However, at 8pm, you may want to hold onto your remote and wait to see the outcome.

In this entry, I will use a simple mathematical model to predict the change in human behavior over time as a function of their sense of urgency.

Partition the game into T discrete time periods (not necessarily equal) during which Batman has a fixed, independent probability Z of intervening successfully. Call the value of life X, and the value of obeying ones conscience Y. (In our game matrix, we will use 0 for clear conscience, and -Y for a guilty conscience.) If batman successfully intervenes with Joker’s plan first, then the game ends, and both ships survive with clear consciences. The game also ends if one ship destroys the other ship, or vice versa, but the surviving ship is left with a guilty conscience. Finally, we will also assume that if both ships flip the switch during the same time interval, then both ships are annihilated, but no moral consequences are felt.

Some Trivial Cases:

Case X > Y, Z = 0. Suppose that batman does not exist, and people do not have a conscience, or a conscience that is worth less than life. Then Joker’s repeated game has a dominant strategy: blow the other ship up as soon as you get the trigger.

Case Y > X, Z = 0. Suppose that batman does not exist, and people have a collective conscience greater than life. If the penalty of violating the conscience exceeds that of dying, then the dominant strategy is to wait until midnight, and let Joker blow them up.

A more interesting case, the game of hope:

Case X > Y, Z > 0. Here batman has a chance Z during each time interval of saving both ships. Hence, at the beginning of the game, the value of the game is a geometric sum from t = 0 to T-1 of Z(1-Z)^t. Provided that batman defeats joker, the resulting utility derived by both ships is X, since both conscience and life are satisfied. Hence, at each time interval t between 0 and T-1, the matrix is given by:

	blow ship	not blow ship
blow ship	0, 0	X -Y, 0
not blow ship	0, X -Y	(1-(1-Z)^(T-t))*X, (1-(1-Z)^(T-t))*X

where the bottom right entry is the expected value derived if both ships decide never to pull the trigger.

When t becomes large enough, the value of not blowing the other ship may become less than X – Y, which leads to a weakly dominant strategy of blowing up the other ship. Note that in the very last interval T-1 (right before midnight), the game reduces to:

	blow ship	not blow ship
blow ship	0, 0	X -Y, 0
not blow ship	0, X -Y	Z*X, Z*X

Hence, if the chance of batman’s intervention is greater than (X-Y)/X, the dominant strategy throughout the entire game is to not blow the other ship. However, if the chance of batman is smaller than (X-Y)/X, somewhere along the way, maybe at 10pm, or 11pm, or 11:30pm, the expected value of not pulling the trigger will become less than pulling the trigger. The best strategy then becomes to blow the other ship up, because it is time to give up hope on Batman.

The game of paranoia

However, this game can be extended once more! Suppose that you also know that the other ship is also reasoning using the same selfish game theoretic principles as you are. You know that at 10pm, both of you are going to give up hope on Batman, and hence blow each other up. In this case, your real deadline is not midnight, but rather 10pm! What happens then? Should you try to blow the other ship up at 9:59pm, knowing that if you do not, then you will die?

However, the other ship also knows that you are thinking this, and hence decides that it would be better to blow you up at 9:58pm. I mean, 1 minute is not enough for Batman to intervene right?

Using this argument recursively, we see that this game becomes one of paranoia, and a dismal game this is! Each ship comes to the conclusion that it is best to press the trigger upon getting it, to minimize the chance of the other ship pressing it first!

The game of mutual trust or cooperation

I do not have a nice rigorous argument for this, but: If there were such a thing as a game of cooperation, it seems likely that the two ships will talk to one another at t=0 and decide to throw out the triggers, even if each ship values life over morality. (Of course, the movie suggests that they did so for moral reasons.) By tossing out the triggers on each ship at time 0, granted that Batman has a sufficiently high probability of intervening, they play a one-shot game based on the expectation that Batman will successfully intervene before midnight. Furthermore, by preventing an extended game, they avoid the terrible equilibrium induced by paranoia.

Of course, because the ships are unable to talk to each other, this decision can only be reached based on some form of “mutual trust” or “faith in humanity”.

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Other notes:

In reality, each human has a different weight attached to life and morality, but here I simplified the problem into a symmetric game assuming that the law of large numbers reduces the variance of each ship’s collective value system, such that the utilities are nearly identical.

Also, Joker could have been lying, just for the heck of it. But such a game would not be meaningful to play, would it?

Great musical comedies

I think it’s about time to revive this blog, although the result may be far less “theory-centric” than it was originally intended to be. But hopefully I will provide more thought-provoking posts from here on out. Before anything substantial though, here are some great links to some great comedic musicians. Igudesman and Joo have the best duets ever!

Mozart Bond

Riverdancing Violinist

Rachmaninoff had BIG hands

Cool and Funny Trailer Remixes

Take your favorite movies and mash them together, or add a different soundtrack, and you have some of the most awesome movie trailers ever. Youtube is the man!

Lord of the Autobots

Scary Mary

Office Thriller

Brokeback to the Future

Titanic Two the Surface

Pimpest Trick Ever

Here’s something you could totally use on teh womens… to get you slapped… but in a way that suggests you are the hottest guy they’ve ever met.

Singshooting

Singshot is a pretty cool site. Record your own voice over karaoke online, and share it with people. =] Even if you hate singing, it’s fun to just browse around and listen to the really good singers. Singshot > American Idol. (Beware though. It’s still buggy.)
Example of some good people:

http://www.singshot.com/playPerformance.html?performanceId=4793&play=Y

Ok, if you don’t like country, here are a couple oldies:

http://www.singshot.com/playPerformance.html?performanceId=15818&play=Y

http://www.singshot.com/playPerformance.html?performanceId=15303&play=Y

Hilarious R Kelly:

http://www.singshot.com/playPerformance.html?performanceId=8372&play=Y 

Weird Al — White and Nerdy

Pwahaha, Weird Al never ceases to amaze me with his creative parodies! Thanks insaneyang.

Really Cool Traffic Paradox!

Did you know that shutting down a road can actually improve traffic in a city?

The phenomenon is known as Braess’ paradox, which states that adding an extra link/road in a network/city could increase overall latency. For a graph theorist, the idea is highly counterintuitive since all the previously existing flows still exist. The reason why this happens, however, is due to the Nash Equilibrium induced by SELFISH drivers. Yes, you know… like… EVERYONE?

Let’s use a simple toy example below (click to enlarge) [1]:

Here, let us assume that all traffic is flowing from s to t along two roads through points v and w. The total traffic is 1 unit (The traffic may consist of many smaller unit flows, e.g. 10 cars contributing 0.1 units of traffic each.). The function l(x) indicates the latency (say, in minutes) induced by x units of traffic along that road; hence in (a), the optimal traffic flow would be to have half the cars choose the top path, and half choose the bottom path. The resulting latency is 1.5 minutes for every driver.
Now, suppose a one-way street was built to connect v to w, as shown in (b). This street is infinitely wide and has no speed limit, and hence cars can essentially “teleport” from v to w with 0 latency. What is the new optimum? Surely, it is at least as good as the old optimum, and in fact it is the same as the old optimum. People could just pretend the one-way Warp 9 road does not exist. However, this will not happen if people do not cooperatively decide to do so. If no driver can communicate (in a friendly way) with other drivers, which is usually the case, the following occurs:

Suppose there are 10 cars. Car #1 sees that it can choose s->v->w->t and make it in 0.2 minutes. Car #2 sees that car #1 has chosen its route, but it sees that s->v->w->t is also beneficial because he can share the road with car #1 and still make it in 0.4 minutes. The same goes for the other 8 cars. In fact, no car will want to choose s->v->t or s->w->t at any time because it will ALWAYS be slower than chooseing s->v->w->t. (They are selfish after all.) Therefore, the result is that everyone chooses the same path, and everyone gets to the destination in 2 minutes.

Hence, adding an extra road causes everybody to lose 30 seconds. WEIRD HUH? So next time you see construction on the road, take a detour and don’t necessary expect a heavier delay than if construction did not exist. You might get to your destination faster!

Such is the price of anarchy.

[1] Tim Roughgarden , Éva Tardos, How bad is selfish routing?, Journal of the ACM (JACM), v.49 n.2, p.236-259, March 2002