Table of Contents

• Introduction

• Every Business Segment Is An Infinite Game

• Oldest Companies In the World

• Oldest Private Equity and Growth Funds

• Other Examples of Infinite Games

• Importance of Long-Term Vision

• Finite vs. Infinite Games in Private Equity

• Private Equity Is A Long Game Business

• Game Theory Principles in Private Equity

• Strategic Adaptation and Flexibility

• The Importance of Cooperation

• Conclusion

Introduction

Private Equity (PE) is often seen as a highly competitive, strategic environment where firms strive to outperform rivals, acquire lucrative deals, and deliver superior returns to investors. Traditionally, it is framed in finite terms: transactions are made, profits are reaped, and deals are closed. However, the application of game theory, especially infinite game theory, offers a more nuanced perspective on how PE firms operate.

Unlike finite games, where players aim to win within set rules and timelines, infinite games are characterized by long-term survival, adaptation, and continuous evolution. In PE, firms play a game where the ultimate goal is not only to "win" but to stay in the game, ensuring long-term success and relevance.

This report explores the "Infinite Game of Private Equity" using key concepts from game theory. It highlights how PE firms maintain relevance, build long-term relationships, manage strategic positioning, and adapt to shifting market dynamics, all while continuing to play the game.

Every Business Segment Is An Infinite Game, And So Is Private Equity

Business, economic activity and trade between people and societies have existed since humans started having surpluses in agricultural and livestock when we stopped being hunter/gatherers.

Businesses existed long before any single company was operating and will exist longer than all the companies that are active today. In any business landscape, rules are flexible and interchangeable, and players come and go.

Additionally, business has no deadline, no end, so there is not an endpoint where we can argue who the winner is. This is why business is an infinite game, so players have to play as infinite players, getting rid of the mindset of necessarily “winning”.

Oldest Companies In the World

The chart above shows the top 20 oldest companies in the world. All these companies have been in business for more than 900 years, and as world history has shown, landscapes have changed strongly since each company was founded. 

Technological disruptions, changes in preferences and trends, macroeconomic conditions, wars, revolutions and more have impacted on the decision-making processes of these companies. We obviously can’t think these companies are doing things the same way they were doing 1,000 years ago. These companies have shown strong resilience and adaptation, following the ultimate goal of an infinite game: Keep playing.

The surprising thing about these companies is that they still did not win, because the game never ended. These companies just keep playing, beyond all the leaders that have come and gone and all the situations they have had to face, their competitors have been entering and exiting the game throughout history.

Oldest Private Equity and Growth Funds in the World

Private Equity is a relatively new space, but as other businesses it is an infinite game too, since it complies with all the conditions to be an infinite game.

The chart above shows the oldest PE and growth funds in the world. Most of these funds have changed management and a big portion of its workforce have had to face changing environments such as like wars, social trends, geopolitical tensions, globalization, and technological disruptions like internet, blockchain, AI and more.

It’s likely when first AI experiments and deployments started with computers playing chess vs professional players, these funds had no idea what AI was, but today all of them are implementing it in its business processes, looking for efficiency and improvements, because they want to keep playing.

A disruption like GenAI, is what we call a “game changer”, but this adjective might be quite wrong. GenAI is “landscape changer”, but the game is still the same: an infinite game, and to keep playing, companies need to adapt and reinvent.

Other Examples of Infinite Games

International Politics

International affairs and geopolitics are prime examples of an infinite game. The pursuit of power on the global stage never truly ends, and nations must keep playing rather than aiming to "win." Few landscapes are as dynamic and unpredictable as the world itself, which serves as the playing field for these nations.

A key example of this game in international politics is the Cold War. When the Soviet Union exited the game, it wasn’t because the United States had won, but because the Soviet economic model depleted its resources, plunging it into an irreversible bureaucratic quagmire and forcing its withdrawal.

This suggests that while the U.S. played the game more effectively, it didn’t defeat the Soviet Union; rather, the Soviet Union chose to leave. Importantly, the game didn’t end there—new players emerged, transforming the once bi-polar structure into a multi-polar one.

Conventional War vs Guerrilla War

In world history we have a lot of examples of two players who think differently about the same conflict. On one side there is a player that thinks about the conflict as a finite game, while its opponent understands it as an infinite game, so tactics deployed by each player are very different.

o A good example is the Vietnam war, where US troops understood it as a finite game with different steps and with a cadence, while Vietnamese were fighting for their lives, their territory and their non-negotiable ideal of a communist society. So Vietnamese troops played to keep playing and to deplete resources and energy of US troops to make them drop out of the game.

o Or Columbia vs FARC: in Colombia, since 1960, the same that happened in Vietnam. Every new government presents a plan to “end” with the FARC guerrilla, thinking it as a finite game, while FARCs have a non-negotiable ideal and plays to keep playing, so the game never ends.

Importance of Long-Term Vision

Having a long term vision is crucial when playing an infinite game because it communicates the values and what the players “fight for”. For example, the United States of America has a long term vision of a free and democratic world, so the US will fight forever, following that vision. When the United States declared independence from the United Kingdom, the founding fathers wrote down the long term roadmap for the nation.

The same happens with companies. The best companies are those that know why and for what they are in this world, and what value they add to society. The best companies do not focus on next quarter’s margin as a strategic goal, they focus on the long term goal of having creditable relationships with their customers to build loyalty. The companies that play the infinite game are those that sell a vision, not a product. For example, Harley-Davidson states that their mission is: “More than building machines, we stand for the timeless pursuit of adventure. Freedom for the soul”. This mission gives the company the guidance to play the infinite game of business.

Finite vs. Infinite Games in Private Equity

In the seminal work "The Infinite Game", Simon Sinek describes two types of games: finite and infinite. In finite games, there are clear winners and losers, defined rules, and an end point. However, infinite games, such as Private Equity, or business in general, are played to continue playing, with flexible and changing rules and a shifting competitive landscape.

• Finite Games in PE: These are the individual transactions—acquiring a company, closing a deal, or exiting an investment. In these contexts, firms might compete with each other for a specific target, fight for higher valuations, or aim for shorter-term returns.

• Infinite Game in PE: The true nature of the Private Equity industry lies in the infinite game—building long-lasting relationships with Limited Partners (LPs), having an important role in economic growth, maintaining a reputation for future deal flows, and adapting to evolving industry standards and regulatory landscapes. The infinite game is about sustaining competitive advantage through continuous learning, innovation, and adaptability.

Private Equity Is A Long Game Business

The average buyout hold period for PE investments in the US and Canada has significantly increased over the past two decades, highlighting the industry's shift toward long-term strategic thinking (a key aspect of playing the infinite game).

From 2000 to 2003, the average hold period fluctuated between 3.6 and 4.7 years. However, starting in 2010, this period consistently extended, reaching an all-time high of 7.1 years by 2023. Globally, a similar trend can be observed, with holding periods increasing from 3.0 years in 2000 to 5.6 years by 2023. This shift underscores how PE firms are adapting to market conditions by prioritizing sustainable growth, continuous portfolio management, and long-term value creation, aligning with the principles of infinite game theory. The average global holding period over this span has risen to 4.4 years, while for the US and Canada, it's 5.2 years, reflecting the evolving nature of deal cycles in the broader PE industry.

Game Theory Principles in Private Equity

Game theory provides a structured framework to understand the strategic decisions made by PE firms in an infinite setting. Some core concepts include:

o Tit-for-Tat and Reciprocity

In an infinite game, reputation is a key asset, and cooperation can yield long-term benefits. PE firms often operate with a tit-for-tat strategy, where they collaborate with peers, investors, and portfolio companies in ways that may seem generous in the short term but build goodwill for future opportunities.

For example:

· Sharing a deal or co-investing with another PE firm may lead to mutual opportunities down the road.

· Fair treatment of portfolio company founders ensures that future entrepreneurs will be more willing to partner with the firm, enhancing deal flow in the long run.

o Iterated Prisoner’s Dilemma

The classic prisoner’s dilemma often reappears in PE deal negotiations. In a one-time game, each player might act selfishly to maximize their gain, but in the context of repeated interactions (the iterated game), the focus shifts to cooperation, knowing that today’s counterpart might be tomorrow’s collaborator. PE firms must carefully weigh short-term gains against long-term reputational costs:

o The Nash Equilibrium in Competitive Bidding

In competitive bidding for companies, a Nash Equilibrium occurs when no firm can unilaterally improve its position by altering its bid, given what its competitors are doing. While many view competitive bidding as a finite game, PE firms playing the infinite game understand the value of not overbidding or chasing every opportunity.

· Bidding wars, driven by the desire to "win" a deal, may lead to overpayment and poor returns—an unsustainable strategy in the long run.

· The infinite player may strategically withdraw from certain competitive bids, ensuring they conserve resources for future deals that align better with long-term goals.

Strategic Adaptation and Flexibility: Staying in the Game

In an infinite game, success depends on the ability to adapt to new circumstances and anticipate changes in the competitive environment. PE firms that thrive understand the necessity of continuous adaptation.

o Adapting to Market Changes

Private Equity has evolved over the years, with firms increasingly incorporating new technologies like GenAI to improve deal sourcing, portfolio management, and due diligence. Firms that do not adopt these innovations risk being left behind. By continuously investing in technology, data analytics, and talent, firms ensure they remain competitive in the infinite game.

o Strategic Partnerships and Alliances

Infinite players understand that the competition today may be an ally tomorrow. Strategic partnerships, syndications, and alliances help PE firms navigate competitive landscapes while minimizing risks. By sharing knowledge, resources, and networks, firms not only improve their current position but also create more opportunities for future collaboration.

For example:

• Cross-border partnerships enable firms to access new markets and diversify risk.

• Collaborating with industry experts helps firms tap into domain-specific knowledge, improving their chances of finding and nurturing successful investments.

Sustainability of Competitive Advantage

A firm’s ability to continue playing the infinite game of Private Equity often comes down to maintaining a sustainable competitive advantage.

o Reputation and Trust

In an industry built on trust and long-term relationships, reputation serves as a cornerstone of a firm’s sustained success. PE firms must work to consistently meet or exceed LP expectations, deliver strong returns, and maintain integrity in their operations.

o Learning and Innovation

The infinite game requires continuous learning. PE firms that invest in knowledge, whether through research, market analysis, or employing experts themselves to adapt to new challenges. Additionally, innovation in investment approaches (such as focusing on impact investing, ESG, or tech-driven growth) can keep firms relevant in the ever-changing market landscape.

Private Equity is a Game that Lends Itself to Cooperation

Even if Private Equity funds must compete to be the best offer in order to become an investor of a good company, that`s a finite game of a few movements, because once the company raised the money and partnered with the chosen PE firm, that game is finished. But in the long run, players need to cooperate to keep playing and choosing a long-term return (as the nature of this business is) instead of a one time and higher return in the next round.

Cooperation is profitable in the long term and economic theory shows that players tend to cooperate more in infinite games than in finite games, and within finite games, players tend to cooperate more when the last round is far away from the current one, and tend to have more conflict and opportunistic behavior when the last round is closer, because the game is close to finish and they want to win.

Math can strongly show what happen when the last round is unknown and it is in the “shadow of the future” (Dal Bo, 2005):

This Equation mathematically shows how continued profits in the long term are better options than one-shot high ticket. In the context of an infinite game, this can be interpreted as follows: since the game is a perpetuity, the probability that it continues (denoted as δ) signifies that there could be many more rounds, and therefore, many more future payoffs. As the discount factor grows, it reduces the present value of the utility of those future payoffs. In simpler terms, it is preferable to have 70 payments of value 65 than one payment of 100 followed by four payments of 35. If a player deviates from cooperative behavior, they may receive an immediate payoff of 100, but will then face punishment, reducing future payoffs to 35.

Relating this to the infinite game of Private Equity, PE firms that focus on short-term gains (such as a large one-time exit) may risk losing long-term value by damaging relationships, reputation, or future deal flow. Instead, firms that prioritize consistent, moderate returns over time—by nurturing long-term relationships, sustaining reputation, and focusing on portfolio management—align themselves better with the infinite game mindset. This approach enables them to stay in the game and secure a steady stream of opportunities and returns, even if those individual returns appear smaller in the short term. Within finance, is a similar approach of stocks valuation, since Value Investing philosophy, represented by legends of the game like Warren Buffet and Charlie Munger (players with both infinite mindset), prioritizes many returns in the long term over a single profit in the present

As Private Equity is an infinite game, cooperation is convenient for all players. But, over the straight Game Theory framework, in this space all the PE funds are partners in search of similar goals in long term.

Conclusion

The infinite game of Private Equity is less about winning individual deals and more about staying in the game for as long as possible, contributing to economic growth by doing smart investments.

Firms that adopt an infinite mindset prioritize relationships, trust, and adaptability over short-term victories. By doing so, they position themselves for long-term success in an ever-evolving, highly competitive industry. Through the lens of game theory, we understand that those who play the infinite game focus on resilience, sustainability, and continuous strategic adaptation—ensuring they remain key players for decades to come.