Basics of Game Theory

Game Theory refers to the mathematical field of study relating to behavior, particularly strategic decision making.

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In Game Theory a “Game is defined by the following characteristics:

  • Multiple Players
  • Payoffs
  • Affected by other player decisions.

This gives game theory a wide range of implications and allows it to be applied in any social interaction where people compete, cooperate, or transact in complex manners. Game Theory has been used to explain everything from Biology, Economy, Politics, Military Strategy, and Psychology.

Usually Game Theory breaks down into two main sides, on which I’ll focus this article on

  • Cooperative
  • Non-Cooperative

However, through this site, I hope to explore the third option in Game Theory concerning complex relationships. Meaning sets of players with changing payoffs that affect multiple games over time.

When discussing cooperative games everyone is working towards a common goal and the issue to be solved is, how much should each player contribute? vs how much should each player profit?

One of the approaches can be attributed to Lloyd Shapley and should answer the following 2 questions:

  1. Is it beneficial to cooperate? Vs going alone? Vs cooperating with other parties instead
  2. How much should I get if I do cooperate?

This is a mathematical approach to fairness according to individual contributions based on the marginal value to the whole and encompasses the following constraints.

  1. Contribution is determined by the value lost or gained by removing their contribution to the whole. (Marginal Contribution)
  2. Players that “Contribute” the same are worth the same, and therefore should be rewarded the same.
  3. If a player’s contribution is zero, their reward from it, should also be zero. (I’ll go further into this topic later, as this may affect relationships, and long-term games)
  4. If a game has multiple parts, each should be analyzed independently from the others

To make this point, I’ll present the following example:

Two business partners (Jack and Jill) are getting together to sell flower arrangements from their gardens.  Individually with the given time and resources Jack could build 100 flower arrangements, while Jill could build 200 flower arrangements. However, working together they could make up to 400 flower arrangements to sell. If they sell each flower arrangement at 10 dollars each, how should the profits be fairly divided?

According to Shapley we have to find the appropriate marginal values first, given all the different orders in which they could be added and then average them together. Let’s start with Jack.

If Jack counted first, his marginal contribution to the whole is 100 arrangements, by this logic Jill’s contribution is 300 arrangements (I know she didn’t make 300, but bear with me, this is all in the name of math).

If we count Jill first, her marginal contribution would be 200 arrangements, therefore Jack’s contribution would also be 200.

So, Jack’s average Shapley Value’s contribution to the 400 flower arrangements would be the average of the first marginal contribution and the second marginal contribution calculated.

(100 +200) / 2 = 150 arrangements out of the 400 total arrangements. This means Jill should get the profits for 250 arrangements. However, for the sake of proof, let’s do the math for Jill (based on already done math for Jack).

In the calculation for Jack, we found that Jill’s contribution was 300 arrangements. In the second calculation we found that Jill’s contribution was 200 arrangements.

(300+200) / 2 = 250.

Math checks out.

In terms of profits IF they decide to cooperate, Jill should make $2,500 dollars (The 250 arrangements multiplied by the $10 Dollars each arrangement costs) and Jack should make $1,500 (The 150 calculated multiplied by the $10 dollars per arrangement).

Back to the original question

Should they cooperate to begin with?

Given that Jack alone could have made 100 arrangements, if he’d sold them for the same price, he would have made $1000 dollars alone, and by cooperating he made $1,500 dollars.

Given that Jill alone could have made 200 arrangements, if she’d sold them for the same price, she would have made $2,000 dollars, by cooperating she can make $2,500 dollars.

Financially, it makes sense for them to cooperate as they all make $500 dollars more when they do, AND IF they use Shapley’s math to divide profits.

Let’s go revisit the example by using a different method, splitting profits evenly (Most common way to split things, and usually considered fair)

Given the arrangements made are the same 400 between the two, and the price stays the same $10 per arrangement. Splitting the profits evenly would give them each $2,000 dollars. For Jack this is amazing, by collaborating with Jill he doubles his expected profits from $1,000 dollars he would have made alone, to $2,000 if he cooperates. However, Jill Could have made $2,000 dollars either way, so she has no financial incentive to cooperate. Jill might have other incentives to cooperate (charity, long term relationship, expected reciprocity) i.e. setting herself up for other games, but that’s not within the current scope of this analysis.

When discussing Non-Cooperative Games, there will be winners and losers

The most well known Non-Cooperative thought experiment is the Prisoners’ Dilemma. There are many different variations of how the story goes, so I’ll borrow the most commonly referenced one via popular social media:

Two members of a criminal gang are arrested and imprisoned (Bubba and Bert). Each prisoner is in solitary confinement with no means of communicating with the other. The prosecutors lack sufficient evidence to convict the pair on the principal charge, but they have enough to convict both on a lesser charge. Simultaneously, the prosecutors offer each prisoner a bargain. Each prisoner is given the opportunity either to betray the other by testifying that the other committed the crime, or to cooperate with the other by remaining silent. The offer is:

If Bubba and Bert each betray the other, each of them serves two years in prison

If Bubba betrays Bert but Bert remains silent, Bubba will be set free and Bert will serve three years in prison (and vice versa)

If Bubba and Bert both remain silent, both of them will only serve one year in prison (on the lesser charge).

Let’s break it down into facts

Players: Bubba and Bert

Payoff: Prison Time

Players affected by the other’s decision? Yes

Here’s a mathematical representation of how it plays out

Bubba and Burt Prisioner Dilemma Image

I suggest you take a few moments to think about the situation.

Game theory suggests that the best thing for you to do is snitch.

Confessing you get the better of the deals on the table, regardless of what your partner does. Let’s look at the partner’s options to understand, remember you don’t know what he does and have no control over it

IF he snitches: The options on the table are either you get 3 years or you get 2 years. Therefore, by snitching yourself, you get the better of those two deals.

IF he doesn’t snitch: The options on the table are 2 years or 0 years. Therefore, by snitching yourself, you get the better of the two deals.

Hence no matter what your partner decides to you, your dominant strategy is to snitch.

And while both keeping quiet, might be the absolute best scenario, it leaves a too much risk open therefore is not a stable solution.

These are just two examples of Game Theory, and how they could be used to solve every day problems.

I’ll admit that these two issues were a bit “Cookie Cutter” but they are meant to explain the basic concepts behind the game theory principles and capabilities. As more articles are published, there will be more and more nuances discovered and addressed. For example, at what point in the Prisoners’ dilemma is it no longer convenient to snitch (Mathematically)? How do major gangs overcome this classic game theory problem (Socially)? At the Shapley’s point, how does it escalate with more and more players? How do each rule and restriction of the theory affect the outcome?

I’ll address some of those issues in my later articles.

In the meantime, if you have an article topic that you’d like me to address personally don’t hesitate to email or leave a comment.

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