Arunabh Sarkar

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What “The Pirate’s Dilemma” in Game Theory Teaches Us About Capitalism

The original “Pirate Game” was popularized by Ian Stewart, an English mathematician

Let’s play a game. Imagine you are a pirate, sailing across the tempestuous Black Sea. You are the leader of a crew which includes yourself and two others. Because the conditions of the sea deter any of you from going on the deck and enjoying the open water, all three of you are stuck inside with a treasure chest full of 100 gold coins that you have stolen from an enemy ship. Now that we have got a bit of a backstory set up, here are the stipulations of the game:

  1. You are the head pirate, which means that you have full autonomy in deciding the distribution of loot. Hypothetically, (don’t worry this will matter later) if you ever died, then the ability to distribute loot would be passed on to your successor.
  2. To determine a successor, the crew has a strict hierarchal structure. To make this more clear, we will refer to you and your crew as agents with different, alphabetical, assignments to make visualizing the hierarchy more simple. You, as the leader, are Agent A, and you do not care too much about the rankings of your crew, so you assign Agent B and Agent C randomly to the remaining crew members. The hierarchy is passed downward if one of you is to die in an epic pirate battle, i.e. if A dies, the head pirate is now B and if B dies the leader (and last pirate standing) is C. [A -> B -> C]
  3. When the head pirate (in this case, you) makes their final decision on how the loot is to be distributed, the rest of the crew (including yourself) must take a vote on if they believe the distribution of the loot is fair. If the majority vote is against the head pirate, they have to walk the plank and the ability to distribute the wealth goes to the successor. In the case of a tie (which is only possible if there is an even number of pirates on the crew), the vote automatically goes to the head pirate's jurisdiction.
  4. You and all the other pirates are rational. Rationality only has two real conditions in this case. First, all the pirates want to live and, second, all the pirates want to maximize the amount of money they get to keep. These are two conditions that will stay consistent and it is important to keep in mind that being rich and being alive are the two most important things to the pirates.

Here is the big question:

What is the most amount of gold that Agent A can keep without risking death?

Take a moment to come up with your own answer before continuing to read. The answer will be revealed down below!

Got your decision captain? Alright, let’s approach this problem by reducing the number of pirates and answering the question in layers.

Say that there is only one pirate (Agent A) and the others have died during the encounter or something. Well, in this case, the answer is simple! Agent A can take all 100 coins and because they couldn’t care less about the deaths of others (yes, even their own crew) they will live a lavish and unbothered life.

Now let us add a new pirate to the scenario. (Agent A and Agent B are present). In this case, Agent A is still the one calling the shots, but now they have another pirate they have to worry about. However, remember what I said earlier; the only thing that the pirates care about is their own wealth and staying alive. Given the stipulation that all votes that end in a tie will end with a ruling in the favor of the leader pirate (Agent A), then Agent A will still take all 100 coins. This fulfills both rational conditions because 100 coins are the maximum number of coins that can be owned, and the vote will end in a tie, ruling in favor of the pirate who decided to hoard the wealth.

Now let us add a third pirate; this is our original scenario. With Agent C involved, the decision-making calculus of Agent A changes a little. If Agent A decides to keep 100 coins, the vote split will be 2–1, in favor of Agents B and C, resulting in the death of Agent A. This is not rational, so Agent A will not do this. Agent A could try to split the coins as close to evenly and perhaps keep their crew happy, but this is not a rational choice, because Agent A cares about their wealth and does not want to give away 67 percent of the treasure. What should Agent A do?

The trick is to buy one agent's vote. Which agent you may ask? Agent C.

The most rational decision for Agent A is to keep 99 coins for themselves, give Agent B nothing and give Agent C a whopping 1 coin. This ensures that Agent A secures the highest possible number of coins while staying alive. Here is why:

Agent A only needs to buy one Agent’s vote to stay alive and prevent the vote split forcing a plank walk. If Agent A is going to buy one Agent’s vote, it has to be Agent C. This is because Agent B’s vote will never be bought rationally. The only scenario in which Agent A stays alive and wins Agent B’s vote is if Agent B keeps 100 coins. This is not rational and therefore is something that the head pirate of a crew would ever do. So, the most rational decision is to buy Agent C’s vote. Agent C’s only opportunity to walk away with some coins in this game is being bribed by Agent A for any amount of treasure. This is because, if Agent A dies from the vote, Agent B will be in charge of the treasure. With only two Agents left in the game, Agent B will keep all 100 coins, and any tie will be in favor of Agent B with Agent C walking away empty-handed. Thus, Agent A can buy Agent C’s vote with virtually any sum of coins. Given that Agent A is still rational and wants to maximize their own wealth, they give 1 coin to Agent C to purchase their vote. This distribution allows Agent A to stay alive and walk away with 99 percent of the profits from the treasure.

Now that we know that rational pirate leaders will always hoard wealth, we can view modern capitalist economies through the lens of the pirates (probably not Agent A though, unless Bill Gates is reading this).

Although the rate of pirate robberies has drastically declined, the rate of wealth inequality has spiked over the course of the productivity boom. Since the 1980s, the Pew Research Center estimates that income inequality has risen 39 percent. As of just a couple of years ago, the richest 1 percent own more wealth than the bottom 90 percent of individuals.

The pirate’s dilemma displays an important dynamic that forms when the wealthy become more powerful and the poor become more helpless. The deadlock between quitting exploitive jobs with little to no safety net disincentivizes workers around the world from taking a stand against the head pirates. In modern capitalist economies, the pirates can be seen analogous to conglomerate industries that have monopolized markets or business magnates who purchase stock buybacks in lieu of raising worker wages.

As Agents get more powerful, it becomes harder for other Agents in the game to reap any benefits. Imagine the pirate's dilemma if there was no vote. Agent A would be given total decision-making authority on wealth accumulated by the efforts of all Agents. With power and wealth skewed in society, game theory demonstrates how individual agents will always act in their own best interest while taking into account the repercussions of other Agent’s behavior. The only time when Agent A cares about other Agents in the game is when there is a potentially negative impact that Agent A themselves will suffer.

It is clear that these games and mathematical models take a far more cynical form in capitalist economies. There are no rules. The rule-makers, which we consider to be the government, are directly influenced by the Agents, or in our case, powerful lobbies, and wealthy donors. When economies operate without rules, they end up helping only a select few (you guessed it, the ones who make the rules). While in-game theory, Agent A faces death if the distribution does not satisfy a single other Agent, in capitalist economies, Agent A faces no repercussions to inequitable distribution. In fact, many times they are rewarded with tax cuts or hefty bail-outs, despite cheating other Agents.

Even more so, consider Agent A who continues to recruit more pirates. The rate of treasure will increase proportionally to the increase in support the crew has to raid more ships. However, the compensation for the rest of the Agents will not change. In fact, the alternating pattern of odd value Agents, like Agent(s) C, E, G, etc., will get 1 coin and the rest will get none. This is because as the pirates get higher returns on treasure, say upwards of 1000 coins for 31 pirates (this math may be off, forgive me). Agent A can still get away with giving one coin to each of the 15 pirates and walk away with 985; a large majority. This proves that as Agents rise and the wealth of the entire crew also rises, the gap is only skewed with the average worker making 0.5 coins and Agent A taking 99 percent of the treasure.

Extrapolating a little from the general Agent analysis that the pirate's dilemma provides in terms of game theory, I propose that the rational and situational conflict between Agents in the pirate's dilemma can be used to explain conflicts between workers in post-industrial revolution industries.

Immigrants have often been the target of conservative politicians who seek to secure base votes or enrage economic-moderates on fiscal policy. A study from the Journal of American Ethnic History showed that immigrants make up 20 percent of the low-wage workforce, 75 percent of immigrants are. professional workers and they also make up 25 percent of the Ph.D. graduates in the United States with all these metrics rising steadily as the world becomes more globalized. However, despite popular sentiment, this does not mean that contributions to the economy are being drained thanks to immigrants. The same study found that undocumented immigrants alone will contribute over $400 billion into the safety net and social security payments of retiring Americans. So, why then, do politicians and wealthy magnates get away with scapegoating immigrants for all of the economic downfalls? Because just like Agent C, E, G, and multiple more, their votes are bought by the hope of achieving the “American Dream”. This is the golden coin that the globe's wealthy Agent A uses to hoard wealth.

The ability to keep America’s economy powerful depends on the backs of domestic work done by immigrants. The Center for Budget and Policy Priorities released a study detailing the significant impact immigrants have on the US economy. They found that immigrants exceed native-born Americans in terms of labor force participation. This hope that immigrants will come to the country where all things are possible and achieve the American Dream is Agent A’s ticket to their vote. It makes Agent B, D, and F all mad. They are not getting paid and they think it is the fault of Agents C, E, or G. The only difference between these two sets of Agents is that Agent A knows it can market one coin to one group and make the other group mad that they are penniless. As a result, the Agents who lack any reward focus on those who were granted an award by Agent A, and Agent A is perceived as deserving of all the wealth because they are the leader, the risk-taker, or some other ludicrous justification for abhorrent levels of wealth.

Wealthy businesses or Agent A control the rules of the game in capitalist economies. This allows for exploitation beyond the initial scenario of the pirate's dilemma. The game teaches us a lot about the consequences of power in singular agents and how, regardless of all the other Agents that follow and the rate at which they grow, more Agents do not mean Agent A is any less powerful.

Unlike the pirate’s dilemma, in capitalism, there are no stipulations, and eventually, it becomes difficult to analyze the status quo through the lens of just a game.