Game theory is a fascinating subject to say the least; yet, quite unknown generally. I was introduced to it back when I was studying economics (I even considered a PhD in game theory) but the reach of game theory spreads way beyond economics, into many fields like biology, sociology, social sciences in general, and of course, our favorite topic: project management!

**What's in here**

- What is game theory, its history and how we got here 🙂
- What is the Prisoner’s dilemma, Nash equilibrium, cool stuff like that!
- How does game theory relate to strategic planning and risk management?
- Examples of situations where game theory is applied in project management

## What is risk management in project management?

When it comes to project management, risk management starts by identifying and analyzing the risks (i.e. uncertain events) involved in a certain project; a plan centered around resources is then put together in order to minimize (or mitigate) those risks. Fundamentally, what you’re trying to do with risk management is decrease the probability, and consequences, of events that could get you in trouble!

### Why use game theory in risk management?

There are many types of concerns we look at when analyzing risks, some of which don’t exactly fall within the sphere of risk management. Consider the conflict arising between two project teams; how is it dealt with? By whom? After all, such a conflict can easily make you lose time, money, and of course, quality!

And so, in project management, we use game theory to model the decision-making process of the many interacting players; project managers, investors, customers, contractors, sub-contractors, or governments. Quite often, their interests may compete, which is why we look at project management scenarios and model them accordingly.

When you combine game theory with risk management, you can manage to capture any opportunity your business has to turn risk into possibility. For managers who deal with a huge amount of data, game theory is simply another way to look at the process of problem solving.

In this article, I’ll be looking at how game theory can help managers solve problems, make strategic decisions and manage risks.

## What is game theory?

Game theory is a branch of mathematics focused on the analysis of strategies in competitive contexts, where the outcome of someone’s choice always depends on the actions of others. Game theory can be applied to a plethora of contexts, of which are politics, war, business and so on.

It’s not an exact science of course; the study of game theory is in fact a study of strategic human interactions; and, the study of human behavior is never really set in stone.

### Who invented game theory?

It’s widely acknowledged that game theory was founded by a Hungarian mathematician John von Neumann, who published his first paper on game theory in 1928. Von Neumann was known in the scientific circle as a mathematical prodigy who could divide two 8-digit numbers in his head by the age of six! There was even a popular saying among the mathematicians of his day: “Most mathematicians prove what they can, von Neumann proves what he wants”.

To be fair though; in 1921, Emile Borel, a French mathematician, published several papers on the theory of games, 7 years before von Neumann. He used poker as the basis for looking into the problem of bluffing and second-guessing one’s opponent in a “game of imperfect information”; Borel was interested in finding out whether every game had a “best" strategy!

Now, he may have been the first mathematician to look into an organized system for playing games, but the thing is, he didn’t develop it very far; which is why most historians give credit to John von Neumann for developing game theory and making it popular!

## What is game theory in project management?

Project management is a great place for game theory to play out; it starts from the very beginning of your project as you negotiate the contract, and throughout, as you manage the project!

The “old” game theory approach would’ve looked at situations where one player does better, at another’s expense, but that’s not very accurate today; instead, we’re looking at a whole range of situations including humans, computers, and so on.

### Are project managers game theorists?

Think about it; what does a project manager do? They manage the connections between projects, they prioritize resources, report progress, and so on. Most of what they do involves dealing with people, negotiating and solving so that the project keeps moving forward.

This is where game theory is as Roger Myerson says “the study of mathematical models of conflict and cooperation between intelligent rational decision-makers”. As a project manager, what you’re doing is setting up and planning your project as a “game” with objectives such as maximizing profit, while you minimize your losses. How do you go about making decisions that aren’t solely based on your existing knowledge, but take into account the decisions of others?

### What are game theory questions project managers should ask?

By asking certain “game theory” questions, and applying the same process to every stakeholder in your project, you can get access to a world of information.

- What am I trying to solve?
- Who are the players?
- What players will have an impact on how successful my decision will be?

## Can game theory be used to model project management scenarios?

There are so many ways in which game theory can be used to model project management scenarios. Take the launch of a product for example; your marketing team has to spend time deciding on a campaign strategy, which takes into account what the competition is doing. Your decisions depend on the decisions of other players who, often, have competing interests.

### Two-player games

Two-player games are used most often to model project management scenarios:

- Government sector / private sector
- Contractor / contractor
- Contractor / subcontractor
- Subcontractor / subcontractor
- Other players

As for the types of games used:

### Cooperative vs. Non-cooperative

Cooperative game theory is a simple approach to analyzing the game at large, without having to make assumptions about who’s in power.

A game is cooperative (coalitional) if players are able to commit in a way that is binding externally; it’s non-cooperative when players fail to form an alliance.

When cooperating, we focus on predicting at the high-level:

- Agreements that will form
- Joint actions taken by groups
- Collective payoffs

Traditional non-cooperative game theory is more focused on predicting the actions of an individual player and analyzing the Nash equilibrium.

### Normal form vs. Extensive form

Normal form games refer to games that are represented in the form of a matrix, or table that shows the players, their strategies, and the payoff. Normal form games basically help identify the strategies adopted by different players, alongside the possible outcomes; the Prisoner’s dilemma is an example of a normal form game.

Extensive form games, on the other hand, are played on trees! Each vertex (node) is a point of choice for a player, while the lines represent possible actions for that player.

To solve an extensive form game, you work things up, back until vertex 1! Meaning that you start with what a rational player would do at the last vertex of your tree (6), then move up to the player at (5) and figure out their moves according to the choices in (6). You then do this all the way up to the first vertex.

### Zero-sum vs. Non-zero-sum

Zero-sum games (constant-sum games) are games in which the available resources can neither increase or decrease, no matter what the players choose. Here, the total benefit for all players in the game always adds up to zero; which means that a player always benefits at the expense of others. Poker and chess are zero-sum games.

The Prisoner’s dilemma is a non-zero-sum game because the outcome has results that aren’t equal to zero; a gain by one player doesn’t have to come at the expense of another.

## What is the Prisoner’s dilemma?

One of the most well-known concepts in game theory, the Prisoner’s dilemma is a paradox in decision analysis; it showcases how strategic thinking between two individuals can produce less-than-optimal outcomes. The Prisoner’s dilemma occurs in any situation where individual decision makers are incentivized to choose a way that isn’t optimal for the group.

### The classic Prisoner’s dilemma goes like this:

Two bank robbers are arrested, after fleeing from the police, and interrogated separately. There are no witnesses, which means that the authorities can only prove the case if one of the robbers betrays their friend.

Each robber has two choices: either cooperate with the police and testify against the other, or remain silent.

- If both robbers “defect” and remain silent, then each of them is convicted of 1 year in jail for running away from the authorities (1+1=2 years total jail time).
- If one testifies but the other doesn’t, the traitor gets no time in jail while their friend gets 5 years (0+5=5 years total jail time)
- If both testify against each other, they each get 3 years in prison (3+3=6 total jail time).

Unfortunately for our robbers, each of them has an incentive to cooperate with the police. If we look at it from Player 1’s point of view:

- If Player 2 remains silent, then Player 1 gets out of here by confessing
- If Player 2 confesses, then Player 1 gets 3 instead of 5 years by also confessing

The paradox lies in the fact that both robbers could minimize their total jail time (1+1=2) by staying silent, yet the incentives they face on their own will always drive them to betray one another, and end up with a maximum jail time possible (3+3=6).

**This is the Nash equilibrium.**

## What is a Nash Equilibrium in game theory?

Nash equilibrium in game theory is a situation in which a player will continue with their chosen strategy (having no incentive to deviate from it), even after taking into consideration the opponent's strategy.

### How do you find the Nash Equilibrium?

To find the Nash equilibrium in any game, you’d have to model out every possible scenario to determine the results, then choose the optimal strategy. In a game of 2 people, you’d have to take into consideration the possible strategies both players could choose. If neither player changes their strategy given all of the information, then a Nash equilibrium happens.

If you'd like to know more about the Nash equilibrium and how it came to be, check out the movie: A Beautiful Mind!

### Why is the Nash Equilibrium important?

The Nash equilibrium helps a player determine the best outcome for them, based on their decisions and the decisions of others. The Nash equilibrium is used in so many places, from business strategies to social sciences, and there’s no specific formula for it. It is determined by looking into different scenarios within the game, and seeing the payoff of each strategy and which would be the optimal one to choose.

### What are the limitations of the Nash Equilibrium?

The main limitation of the Nash equilibrium is that it requires an individual to know the strategy of their opponent. But a person rarely knows their opponent's strategy or the outcome they want, which is why the Nash equilibrium doesn't always lead to the most optimal outcome; it just means that someone is choosing the best strategy for them based on the information they have. Basically, a player will continue with their chosen strategy, with no incentive to change course, such as in the Prisoner’s dilemma example above.

### What is a real life example of the Prisoner’s dilemma?

Say, we’ve got a duopoly where two strong companies dominate the market. Each of these companies will own a certain section of the market where they’ll behave like a monopoly. If there’s a recession for example, these monopolistic markets will likely end up merging; suddenly there will be one market for the product, not two.

**Should each of these companies accept the new context, or should they push each other out? **

If the value of the average costs of both companies is higher than the value of the average costs of one of them, then the total costs would be lower if the market became monopolistic; a merger could be attractive then.

On the other hand, if both companies decide to get into a price war and try to push each other out of the market, the following situation can arise: both companies will fail to maximize profits right away by decreasing the price (that's for sure), but the company that manages to “attack” first will end up acquiring a decent portion of the competitor’s customers. The competitor will then have no choice but to bring down their price; yet the price can never go below cost. 🙃

## What are real game theory uses in risk management?

Game theory can be used extensively in risk management; here are a couple of examples.

### You wish to increase your market share

If you lower your prices, how will competition respond to you? Would you have launched a price war? Will your customers show loyalty or will they switch because of some new perception they have of you and your product? Will your investors get scared? Are you able to maintain low prices for some time until you drive a few competitors out of the market?

### You’re unable to produce all the features your customers want

You’ve got a team that can only build 3 new features in the next 5 months, but your customers have asked for 7. What will happen to them if you don’t succeed? And how about your competitors? Will they take advantage and build the features you’re unable to tackle at the moment? Will focusing on these 3 new features give you a lead in innovation, which will then give you some market share? How will your investors respond? Are you able to hire more people to produce the 4 other features your customers would like to see? Is it better to have all 7 features ready together, even if it means delaying the launch of your new product version?

As you can see, there are many things to consider when tackling risk management; and every option chosen represents a set of risks and opportunities. Assessing these risks is no simple task, the process can vary widely across different contexts; the assessment of risk will change based on the point of view: CEO, Middle management, developers, and so on.

## What's the bottom line?

In a typical risk assessment approach, the probabilities are usually guessed; there isn’t much guidance on how to get them right! Actually, it’s easier to get data on how stakeholders assess the values of outcomes.

By using game theory however, you derive your probabilities from real analysis of opponent strategy, which then becomes the basis for your mitigation measures. Game theory models that are clear about the nature of interacting decisions, with distinguished choices and random variables, can bring about more effective risk management recommendations.

Risk analysis and game theory cannot (or should not) be separated, but rather allowed to complement one another. At least, that’s what the experts think! 🙃