Mental Models (Part 4)

Use these mental models to improve your problem-solving and decision-making skills and overcome common [smoov.bra.in2023] reasoning errors. 

PDF: We created a clean, printable PDF of this series for offline reading. Buy it here or view a preview.

Greetings, my fellow plebeians.

I hope you are enjoying this series so far. 

In case you missed the first three installments, you can check them out here:

Mental Models: Part 1 | Part 2 | Part 3

And if you’re keeping score at home, you’ll know we have discussed 9 Mental Models so far. 

Today is the grand finale, number 10.

But you don’t have to stop at just ten.  

There are thousands of mental models out there that you can study (it would take years for me to cover them all) that can help you avoid [smoov.bra.in2023] reasoning errors and suboptimal outcomes. 

This series is just an intro to get you started.

Think of it as a starter-kit overview of the ‘Big Models’ that you need to master in order to gain a basic understanding of how the world works, how to manage uncertainty and risk, and how to make better decisions. 

It was designed to highlight some of the most important models that apply broadly to life and are useful in a wide range of situations.

So, today we’re going to talk, discuss, and analyze Game Theory, which is quite arguably, the most important model of the series so far. 

This will be the final chapter in our Mental Models series, so, let’s pick up where we left off in Part 3 and continue our journey into the mind-bending (err I mean super awesome) realm of mental models, where the horizon of understanding stretches endlessly before us.

And now, here is Part 4:

10. Game Theory

“Don’t ever play games with me,” she said.

“Everything is a game, baby,” he said.

In today’s multifarious world of free enterprise and big data, the relentless pursuit of strategic advantage has compelled modern enterprises (from tech giants to financial institutions) to lean on heavy analytical frameworks to decode complex interactions, probabilities, and decisions.

One such framework that has transcended from the dark realm of theoretical economics to widespread mainstream applicability in various sectors is the Theory of Games.

Or, better known in modern times as: Game Theory.

So, what is Game Theory?

Game theory, at its core, is the study of mathematical models of strategic interaction among rational decision-makers.

Or, in other words: 

It’s the science of strategies which come under the probability distribution. It determines logical as well as mathematical actions that should be taken by the players to obtain the best possible outcomes for themselves in a game.

The focus of game theory is the game, which serves as a model of an interactive situation among rational players.

The key to game theory is that one player’s payoff is contingent on the strategy implemented by the other player.

Game Theory operates around three principal components: Players, Strategies, and Payoffs.

• Players: The decision-making entities in the game.
• Strategies: A comprehensive action plan under different scenarios.
• Payoffs: The outcomes for different combinations of strategies. 

Game Theory essentially involves the strategic interaction (strategy) between two or more participants (players) that results in a set of circumstances (the game) — arriving at either a good or bad outcome for the players (the payoff).

It consists of the following assumptions:

1. All players know the rules of the game.
2. All players are rational decision makers.
3. All players will strive to maximize their payoffs in the game (acting according to their personal self-interest).

Initially founded as a tool for economic modeling by Von Neumann & Morgenstern in 1944, game theory has found uses in various sectors including, evolutionary biology, war, politics, psychology, engineering, and business.

It also has potential applications in your personal life and in the workplace. 

In his book “Game Theory“, Morton Davis describes it as following:

“The theory of games is a theory of decision making. It considers how one should make decisions and to a lesser extent, how one does make them. You make a number of decisions every day. Some involve deep thought, while others are almost automatic. Your decisions are linked to your goals—if you know the consequences of each of your options, the solution is easy. Decide where you want to be and choose the path that takes you there. When you enter an elevator with a particular floor in mind (your goal), you push the button (one of your choices) that corresponds to your floor. Building a bridge involves more complex decisions but, to a competent engineer, is no different in principle. The engineer calculates the greatest load the bridge is expected to bear and designs a bridge to withstand it. When chance plays a role, however, decisions are harder to make. … Game theory was designed as a decision-making tool to be used in more complex situations, situations in which chance and your choice are not the only factors operating. … (Game theory problems) differ from the problems described earlier—building a bridge and installing telephones—in one essential respect: While decision makers are trying to manipulate their environment, their environment is trying to manipulate them. A store owner who lowers her price to gain a larger share of the market must know that her competitors will react in kind. … Because everyone’s strategy affects the outcome, a player must worry about what everyone else does and knows that everyone else is worrying about him or her.”

When employed as a mental model, Game Theory provides simplified structures to dissect the chaotic web of interactive decisions. It can give you (and your company) the ability to streamline choices and predict how other parties will react in given situations. 

So, now that you understand the basic premise of the theory, now you’re probably wondering what is meant by “games” and how these games are played.  

Taking a page from Game Theory and Strategy:

Game theory is the logical analysis of situations of conflict and cooperation.

The “game” is a waltz between conflict and cooperation. 

More specifically, a game is defined to be any situation in which:

1. There are at least two players. A player may be an individual, but it may also be a more general entity like a company, a nation, or even a biological species.
2. Each player has a number of possible strategies, courses of action which he or she may choose to follow.
3. The strategies chosen by each player determine the outcome of the game.
4. Associated to each possible outcome of the game is a collection of numerical payoffs, one to each player. These payoffs represent the value of the outcome to the different players.

So, in simple terms, you can define a game as a model that represents a situation in which multiple agents (called players) make decisions that result in outcomes with payoffs for each player.

Game theory is the study of how those players should rationally play games. 

At the end of the game, each player would like the result to be an outcome which gives him as large a payoff as possible.

AKA: You play to win the game. 

So, what kind of games do players play?

That’s a great question, I’m glad you asked that question. 

Let us first cover some basic lingo. 

Games in matrix form can only represent situations where people move simultaneously. 

Within games of this nature, we typically see the following params:

1. The sequential nature of decision making is suppressed.
2. The concept of ‘time’ plays no role.

These are called simultaneous games. These are games where the decisions of players are simultaneous: both you and the other ‘player’ choose at the same time. The simplest example of this type of game is probably ‘rock, paper, scissors’.

But many situations involve players choosing actions sequentially (over time), rather than simultaneously. 

These are called sequential games. 

Sequential games are strategic interactions where players make decisions in turns, knowing the moves of those who acted before them.

Chess is a good example of a sequential game. 

Sequential games are commonly represented using extensive-form game trees (we will discuss trees a bit later) which capture the sequence of actions and associated payoffs.

Within a sequential game, solution concepts like backward induction and subgame-perfect Nash equilibrium are used to find optimal strategies for each player.

This means that:

• Players who move first can significantly influence the game.
• Players who move later on in the game have additional information about the actions and reactions of other players.
• Players make their next action conditionally based on additional information received during the game.

Note for the nerds: a Subgame Perfect Equilibrium (SPE) is a Nash equilibrium with the property that all players play best responses after each history of the game. You can solve for SPE by using backward induction. 

Sequential games are a form of the non-cooperative game theory, which if you’re just starting out, should be the first game theoretical concept you try to master.

Potential Landmine: while sequential games allow for better prediction and planning that can lead to better decision making, they tend to ignore the extremely dynamic conditions of strategic situations. 

That said, non-cooperative games have been associated with a multitude of challenges and difficulties (just ask any business school or MBA student) and a prime example of this (that is taught in lectures all over the world) is a concept many scholars like to call Prisoner’s Dilemma. 

The Nuances of Prisoner’s Dilemma:

Let’s imagine a fictional story using two characters, Jack and Oliver, childhood friends turned professional criminals, who meticulously planned a jewelry heist that involved robbing a high-security vault storing diamonds worth millions of dollars.

Their plan was perfect, and the heist went off without a hitch, or so they thought.

Not even an hour after the robbery, they found themselves surrounded by police and were subsequently arrested.

The police didn’t have enough evidence to convict them for the heist itself (the diamonds were hidden securely) but they were caught with some burglary tools (not definitely incriminating but definitely suspicious) and were suspects only because a partial license plate match was captured by a traffic camera near the heist location. A weak case at best. But, if either of them talked and provided evidence against the other, that would be a different story altogether.

They were both separated and placed in individual interrogation rooms, a thick wall of silence between them.

The Interrogation and Offer

The detectives presented each suspect with the same proposition: The evidence for the heist was circumstantial; but, if either of them confessed and testified against the other, that person would get full immunity.

But full immunity comes at a price, you will walk free, but your buddy would take the fall and face 15 years in prison.

If both suspects stay silent, however, the police would only be able to book them on minor charges, resulting in 1-year sentences for each.

And it they both confessed, each would serve 5 years.

The Dilemma

The situation can be presented using a traditional payoff matrix:

	Oliver Silent (Cooperate)	Oliver Confesses (Defect)
Jack Silent	-1, -1	-15, 0
Jack Confesses	0, -15	-5, -5

In this example, the first number in each cell represents Jack’s sentence in years, and the second number represents Oliver’s sentence.

• Both Silent (Cooperate/Cooperate): 1 year each.
• One Confesses, One Silent (Defect/Cooperate): 15 years for the one who remains silent, 0 years for the one who confesses.
• Both Confess (Defect/Defect): 5 years each.

Jack’s Calculations: Jack (let’s assume he is a risk-averse individual) begins to calculate his best strategy. If his buddy Oliver remains silent and they both get 1 year, that would be the best collective outcome. But, on the other hand, the lure of walking away free is strong. Plus, if Oliver confesses and Jack remains silent, he’s looking at 15 years. This terrifying possibility makes Jack strongly consider confessing, as it minimizes his worst-case scenario to 5 years. What to do?

Oliver’s Calculations: Oliver (let’s assume he is more of a gambler), thinks along similar lines but is more optimistic that Jack will remain silent. Afterall, Jack is his ride-or-die since childhood, right? Oliver sees the mutual benefit of both serving only 1 year but can’t completely ignore the risk of Jack betraying him. That is a real possibility. Who can you trust when it comes to time in jail? Usually, you can’t trust anyone. So, like Jack, he leans toward confessing to hedge against the worst-case scenario of receiving a 15-year sentence.

So, how do you break the dilemma? 

The dilemma intensifies as both realize their optimal individual strategies (confessing) conflict with the optimal group strategy (staying silent). If either had a way to credibly commit to staying silent, both would benefit. Unfortunately, self-interest and the inability to communicate make this situation very difficult (and stressful). 

The tipping point occurs when both players recall their childhood pact to never betray each other. Though it was made in youthful innocence, it now bears the weight of very adult consequences. So, both players, both longtime buds, now consider this pact and wonder if it’s strong enough to withstand the pressure and high stakes of the situation. 

As time runs out, they both are forced to make their decisions. Drawing upon their friendship and the realization that betrayal would irrevocably damage something invaluable, they both choose to stay silent. When the detectives return to collect their decisions, they are disappointed, realizing the strongest weapon they had was the uncertainty each suspect had about each other’s decisions.

In the end: Jack and Oliver each receive 1-year sentences on minor charges, a far cry from the 15 years they risked or even the 5 years had they both betrayed each other. Their friendship remains intact, and they serve their time knowing they faced the Prisoner’s Dilemma and emerged victorious as a unit. Perhaps, they will also return to the spot where they stashed the treasure after they served their sentences and find it is still there, untouched. 

By now you are probably beginning to see the irony of this dilemma. 

If Jack and/or Oliver act selfishly and do not cooperate (one of them rats out the other), they will be worse off than if they were to act unselfishly and cooperate together (both of them stay silent).

Table 1: The Payoff Matrix

	Oliver Silent (Cooperate)	Oliver Confesses (Defect)
Jack Silent	-1, -1	-15, 0
Jack Confesses	0, -15	-5, -5

Table 2: Individual vs Collective Payoffs

Strategy	Jack's Payoff	Oliver's Payoff	Collective Payoff
Both Silent	-1	-1	-2
One Confesses	0 or -15	0 or -15	0 or -30
Both Confess	-5	-5	-10

By understanding these numbers, it becomes evident that although individual rationality pushes each towards betrayal, the collective rationality suggests cooperation as the mutually beneficial strategy.

This dichotomy lies at the core of the Prisoner’s Dilemma, serving as a model for various scenarios in economics, politics, and even in interpersonal relationships where trust and the temptation to betray compete.

Some real-life examples include: two countries competing in an arms race, businesses engaging in a price war, farmers increasing their crop productions, a text from a lover that says “if you really loved me…”

You get the idea. 

So now we will briefly go back to where we started and tie a few things together: 

“Don’t ever play games with me,” she said.

“Everything is a game, baby,” he said.

In a marriage, for example, the division of household chores can be framed as a classic Prisoner’s Dilemma game. Both partners benefit most when they both contribute, but each has an individual incentive to be lazy and get a free ride on the other person’s effort.

Within this game, mutual cooperation maximizes joint happiness, but individual incentives may lead to suboptimal outcomes. And it is a game where both parties benefit from mutual investment but risk emotional loss if the other doesn’t reciprocate.

The challenge here lies in aligning individual incentives with collective well-being to reach an optimal, cooperative equilibrium, and many of these situations, equilibrium is never reached. 

Many such cases. 

So (in theory) for the best outcome(s) to happen, players should cooperate.

This leads us into another theory that’s called the Theory of Moves.

According to the Theory of Moves, shifts in outcomes can be largely predicted on how the play starts.

We will come back to this in a moment. 

But first, let’s define what the Theory of Moves is:

The Theory of Moves, developed by economist Steven J. Brams, is a dynamic extension of classical game theory that considers the sequential decision-making of players. It assumes players think ahead and anticipate the consequences of their moves and the countermoves this will induce from other players.

In this framework, players continuously adapt their strategies over time, instead of making a one-shot decision. The goal is to identify stable states where no player has an incentive to unilaterally deviate or change their strategy, given the subsequent adjustments and reactions of the other players. 

Now, let’s circle back and apply it to the Jewelry Heist Scenario from above: 

In the bestselling book “Thinking Strategically“, Dixit and Nalebuff may have put it best: “Everyone’s best choice depends on what others are going to do, whether it’s going to war or maneuvering in a traffic jam. These situations, in which people’s choices depend on the behavior or the choices of other people, are the ones that usually don’t permit any simple summation. Rather we have to look at the system of interaction.”

Something to ponder: are most people willing to choose a death they are familiar and more comfortable with than risk the unknown?

Some Additional Game Theory Applications:

Strategic Choices in Space Alliances

As more private companies enter the domains of space exploration and galactic warfare, they face crucial decisions concerning partnerships and resource allocation. For example, Deimos-One could form a strategic alliance with another tech giant to co-develop a new stratospheric platform for detecting and transporting Moon ice.

Game Theory can be utilized to analyze whether forming such an alliance would be mutually beneficial or if going solo would yield higher payoffs.

Table 7: Strategic Choices in Space Alliance

Decision	Deimos-One Payoff	Tech Giant Payoff	Combined Payoff
Form Strategic Alliance	80	75	155
Go Solo	70	65	135

Here, the combined payoff of forming an alliance is 155, which is higher than going solo. Thus, Game Theory would suggest that the alliance is the optimal decision for both parties.

How a firm interacts with other firms plays an important role in shaping sustainable value creation. Here we not only consider how many companies interact with their competitors, but how companies can co-evolve. Game Theory is one of the best tools to understand interaction. Game Theory forces managers to put themselves in the shoes of other players rather than viewing games solely from their own perspective. The classic two-player example of game theory is the prisoners’ dilemma. — Michael J. Mauboussin

Final Thoughts

If you have made it this far, you have made it to the end of yet another long-boring analysis on Game Theory and subgame perfection.

For the sake of being boring, however, let us consider, is there a Nash Equilibrium in total global annihilation?

World War 3 is in fact, just around the corner. 

So, just for the hell of it, let us consider the scenario of total global annihilation in high stakes war games.  

Nuclear Deterrence and Mutually Assured Destruction

At the time of this writing, tweeted under my hand, this 27th day of October, anno Domini 2023; where we sit Neanderthal on one hand and Singularity on the other, where any such event of significance and power (nuclear or otherwise) may catapult us forward into the future or a thousand years back into the dark ages, no man yet has ever possessed the knowledge to know things unseen, but often, will speak prophecy about the end according to his taste.

I work in probabilities, but I cannot 100% accurately predict the future.

No one can. 

The reason for this is simple: it’s due to the paradox of prophetic brilliance, which means the fool must be intelligent enough to recognize that he is, in fact, a fool.

It is in a sense, a meta-layer of ignorance — the ignorance of your own ignorance.

A massive contradiction.

And yet, here we sit. 

All fools. 

Strategically posturing in a bipolar world.

Many such cases. 

One of the most cited examples of this strategic posturing is the Cold War standoff between the United States and the Soviet Union. In this two-player game, the strategic posture essentially boiled down to “mutually assured destruction” (MAD). The game examined the benefits and risks of pre-emptive strikes versus a strategy of restraint and deterrence.

Table 8: Nuclear Deterrence Payoffs

Decision	USA Payoff	USSR Payoff	Combined Payoff
Both Deter	-10	-10	-20
One Pre-empts	-100	-5	-105
Both Pre-empt	-100	-100	-200

In an obscurantist world with imperfect information, where not all actors are rational, where rational actors can have lapses in judgment, and where tension and emotion can reach apogee, it can be incredibly difficult to evaluate risks and predict outcomes properly and/or effectively.

In this case, the payoffs represent the catastrophic consequences of nuclear war. Both nations face an incentive to deter rather than pre-empt, resulting in a Nash equilibrium of mutual deterrence.

That said, given the infinitely stochastic nature of our planet (a realm devoid of consistency or asymptotic normality), it can be quite difficult to accurately predict the future and make good decisions given the: (1) intricate interplay of the variables (known and unknown); (2) constraints of our inherent cognitive biases which limit our predictive abilities; (3) natural psychological drive to overestimate the likelihood of catastrophic events due to availability heuristic/fear/sensationalism; (4) fact that many phenomena of the cosmos (on Earth and elsewhere in the universe) remain mysterious and beyond our current comprehension.

To add to the complexity and uncertainty, humans are a species unlike any other in the known universe. 

Their behavior is not random. 

Human behavior is systematic and predictable — making the species predictably irrational.

This may destabilize the system away from its Nash equilibrium.

To add to the complexity and uncertainty, the humans are a war-like species (constantly locked in never-ending battles over scarce resources) so their home planet is constantly under threat of annihilation (whether self-inflicted or otherwise), adding additional layers and dark tunnels to your reasoning path.

You see, on Earth (Solar System 1) the future is inherently unpredictable because not all variables can be known — and even the tiniest error in your analysis can quickly throw off your predictions.

On Earth, the home of irrationality and stochasticity, chaos and uncertainty rule the day. 

And because the future on Earth is not 100% deterministic, decision makers need to find ways to shine light on the reasoning path, to help navigate through the dark tunnels of uncertainty and complexity and gain a better understanding of the events that could have a significant impact on a multitude of potential futures, whether positive or negative. 

This is where your newfound knowledge of game theory will come in handy. 

In times of uncertainty, game theory should come to the forefront as not only a strategic tool, but also as a mental model to navigate complexity in high stakes decision making situations. 

Sure, there are some who consider game theory to be more theoretical than practical.

But many of them are simply academically unable to convert theory to application.

Their models are sloppy.

Their implementations are very linear.

And their strategy is designed to offer a single, overly precise answer to a very complex problem. 

The results are usually subpar. 

A common [smoov.bra.in2023] runtime error.

The key, I have found, to building a working model that is applicable in real-life situations is to (1) develop a range of outcomes (assuming your players are rational); and (2) map out the advantages and disadvantages of each option.

This way, you can avoid “shitty analysis syndrome” where you get a binary “yes or no” to a situation that is not so black and white.

This often requires you to develop a comprehensive and strategic framework to support your decision-making process, which will take you much deeper into the decision tree. 

User Advisory: developing a working model that extends beyond a useless theoretical analysis is intended for seasoned vets and those willing to go deep into the pain cave. User discretion is advised. 

One of the worst things I have observed (especially in the corporate setting) is that uncertainty can paralyze decision making, and, perhaps worse, compel managers and stakeholders to base their actions on gut feelings and not much else. 

These “gut feelings” better known as intuition bias; will get you killed in high stakes games. 

But, a working game theoretical model, however, can provide clear information to the decision-making process, but only if it is modeled in a way where the inputs are detailed enough to make the methodology practical and a wide range of probable scenarios can be properly analyzed.

This can generate good results, even in complex environments. 

In our Lab, we have applied these types of models to many types of environments, with positive results. 

In our Earth observation platform, for example, we have examined the dynamics of airspace control with adversarial elements in near space environments. In particular, scenarios where our HALO vehicle is at risk of being jammed or interfered with by adversarial airships or drones.

Noting that downside risk needed to be minimized, we used the Minimax equilibrium to find optimal paths that could curtail the worst-case scenario for the airship. This is crucial for mission-critical applications where the downside risk needs to be minimized.

We also looked at strategic options for the AI powered prioritization of specific observational tasks using a real-time auction mechanism, especially when the airship is battling high-speed winds. Each task had different priorities and requirements, and HALO needed to be able to allocate its limited propulsion energy efficiently while contending with 275+ mph (239 kts / 442 km/h) jet stream winds.

This was modeled as a Vickrey-Clarke-Groves (VCG) mechanism where each payload on board would bid for priority in observational tasks. HALO’s AI acted as an ‘auctioneer’ and allocated propulsion energy based on the bids and stochastic elements in near space (e.g., wind speed).

Pareto optimality can also be applied here. A Pareto-optimal solution would allocate energy in such a way that no single task can improve its observational quality without adversely affecting another task. The AI was used to find such Pareto-optimal solutions in real-time.

The energy allocation was then adjusted dynamically based on real-time wind data and state of each observational task. This ensured that the most critical tasks were prioritized when wind conditions were most challenging, thereby maximizing the overall value of the observations. 

Both of these scenarios were modeled using a game-theoretic approach, allowing us to achieve multiple goals, even in stochastic environments. 

But like I mentioned earlier, game theory is just a tool, and a tool is only as good as the person using it. These models are usually only helpful if company decision makers can extract information that can help them make informed decisions based on a wide range of actions by the players (or competitors) involved. 

That said, when you are starting out building your own model, or trying to figure out the proper solution steps for a challenging problem, it can help to break down the complexity into basic elements such as: 

• Are the players in this game rational?
• Are the players acting according to their self-interest?
• Do the players all understand the rules of the game?
• Can we reach Nash equilibrium?

If you cannot meet these parameters, you should determine options to optimize for: rationality, self-interest, game rules, Nash. 

Within those key variables, you should be able to determine the necessary tradeoffs and optimizations required to improve your model and shift it to be more aligned to make data-driven (rational) decisions. 

Well, I think I’ve rambled on enough for today.

I hope you gained some valuable insight from this series and will leave just a bit smarter than you were when you started. 

So, what Mental Model has helped you the most?

Let me know on X/Twitter.

Follow me for more shitty analysis: twitter.com/jaminthompson.

Mental Models: Part 1 | Part 2 | Part 3

Buy The PDF

Best Mental Model Books for further study: CLICK HERE