Shield Academy | Introduction to non-cooperative game networks and the Nash Equilibrium

As human beings, our existence has always revolved around surviving in a highly competitive world. When there are limited resources, constraints between two or more parties are unavoidable and natural. Typically, when faced with restrictions in a win-or-lose situation, competition almost always arises. And the interactions between species that arise as a result of such competitions may be viewed as a multi-player game. This is where game theory enters the equation.

For the uninitiated, game theory is an area of applied mathematics used to develop the most effective strategy for achieving success in various competitive scenarios. In essence, game theory deals with the various strategic interactions between multiple decision-makers, commonly known as players or agents.

Although ‘The Theory of Games and Economic Behavior’ (1944) by John von Neumann and Oskar Morgenstern is often credited with launching the field, game theory has evolved significantly since then, both in terms of the number of theoretical findings and the extent and variety of applications. Since 1970, the Nobel Prize in Economic Sciences has been awarded to a total of 12 prominent economists and scientists for their contributions to game theory, with John Harsanyi, John Nash, and Reinhard Selten receiving the first such honor in 1994 for their “pioneering analysis of equilibria in the theory of non-cooperative games.”

This clearly demonstrates the relevance of game theory in modern-day analysis and decision-making. Although von Neumann and Morgenstern’s 1944 work is widely considered to be the origin of the scientific approach to game theory, game-theoretic concepts and certain isolated significant discoveries also came from earlier years. While it is unclear which exact event or writing was the catalyst for game-theoretic thinking or decision-making, it is undeniable that the second half of the twentieth century was a golden era for game theory. And the twenty-first century is set to be a platinum era, as it will unravel the various theorems and applications of static and dynamic games.

From business, finance, and economics to political science, psychology, biology, military, and more, game theory can be applied to various industries. And having a solid understanding of game theory strategies plays a key role in enhancing one’s reasoning and decision-making skills in today’s highly complex and competitive world.

Different types of games are used in game theory to analyze different types of situations. Some of the prominent factors that influence the type of games developed include the number of participants, symmetry, and player collaboration. Two of the most popular games that come under game theory include cooperative and non-cooperative games.

An introduction to non-cooperative game theory

In a non-trivial game, a player’s objective function is usually influenced by the choices of at least one other player, if not all of them. As a result, a player cannot simply maximize their own objective function without regard to the other players’ decisions. This essentially establishes a connection between the players’ actions and brings them together in decision-making, resulting in the field of cooperative game theory, which deals with issues like bargaining, coalition formation, and the allocation of excess utility, among other things.

However, if the players are not allowed to cooperate, it results in non-cooperative game theory. According to game theory, a non-cooperative game is one in which individual players compete rather than cooperate, and alliances may only work if they are self-enforcing. Non-cooperative games are often studied using the non-cooperative game theory framework, which aims to anticipate players’ individual tactics and payoffs and discover Nash equilibria.

What is the Nash equilibrium?

Named after mathematician John Forbes Nash Jr., Nash equilibrium is the most prevalent approach to define the solution of a non-cooperative game that consists of two or more players. The best result of a game in a Nash equilibrium is one in which no player has an incentive to stray from their originally selected strategy after considering an opponent’s decision. This means that each player is expected to be aware of the other players’ equilibrium tactics, and no player stands to benefit from modifying only their own strategy.

The principle of Nash equilibrium, which is considered one of the most fundamental principles in game theory, can be traced back to Antoine Augustin Cournot, a French philosopher and mathematician who applied it to competing companies determining outputs. The fundamental concept of the Nash equilibrium is based on the idea that analyzing individual decisions does not allow one to anticipate the choices of other decision-makers. Instead, one must consider what each player would do in light of what the others are likely to do. Given that neither player wants to reverse their decision based on what the others are doing, the Nash equilibrium necessitates consistency in their decisions.

If a game has a unique Nash equilibrium, all rational players should converge to the state represented by the equilibrium, i.e., each player should try to adopt the strategy that maximizes its utility function. This isn’t to say that every game will have a Nash equilibrium. Some games have just one Nash equilibrium, some have none, whereas others have multiple Nash equilibriums.

Examples of Nash equilibrium

Nash equilibrium is applied in a wide range of situations, including hostile situations such as wars and arms races and conflict minimization or resolution through repetitive interaction. It is also used to analyze economic aspects like markets, currencies, and auctions. The Nash equilibrium is also used to examine how effectively people with different tastes can collaborate and whether they’re ready to take risks to get a cooperative outcome.

A prominent example of the Nash equilibrium in game theory is Prisoner’s dilemma, a concept developed by Merrill Flood and Melvin Dresher in 1950. A prisoner’s dilemma is a circumstance in which two players acting strategically make a poor choice for both of them. According to the Prisoner’s dilemma theory, it is up to the two parties to decide whether or not to cooperate. Regardless of the other party’s decision, any party is offered the opportunity to defect. Without a doubt, individual collaboration is required to make better economic decisions. And the Prisoner’s dilemma clearly demonstrates that when each person follows their own self-interest, the outcome is poorer than if they cooperated.

How does the Nash equilibrium work?

The Nash equilibrium is basically a mathematical and logical attempt to identify participants’ actions to achieve the greatest possible results for themselves.

Let us look at an example to understand this concept better. Say there are two companies Alpha and Beta. Let’s assume that Alpha and Beta chose strategies A and B, respectively.

  • Then (A, B) is a Nash equilibrium if Alpha has no other strategy that maximizes their reward better than A in reaction to Beta selecting B, and Beta has no other strategy that maximizes their reward better than B in response to Alpha choosing A.

Let us now assume that there are two more companies, Gamma and Delta, who chose strategies C and D, respectively.

  • Then, if A is Alpha’s best reaction to (B, C, D), B is Beta’s best response to (A, C, D), C is Gamma’s best response to (A, B, D), and D is Delta’s best response to (A, B, C), then we can say that (A, B, C, D) a Nash equilibrium.

Let us illustrate the above through a real-world scenario. Assuming Alpha and Beta are two individual entities that compete with each other, let’s say that both the entities are thinking of launching a new marketing campaign through social media.

  • If both Alpha and Beta follow through with the campaign, both entities will gain 250 customers each.
  • If only one entity follows through with the campaign, that entity will gain 500 customers, while the other gains none.
  • If neither entity decides to execute the campaign, neither will attract new customers.

As a result, Alpha should carry out the social media campaign since it offers a greater return on investment than not doing so. Beta should also do the same. Therefore, the situation in which both firms execute their social media campaigns is called a Nash equilibrium. One of the most prominent features of a Nash equilibrium is that it is stable, in the sense that if two distinct players begin playing at the Nash equilibrium, neither of them will benefit from unilateral deviation from these rules.

How to find the Nash equilibrium?

In order to identify the Nash equilibrium in a game, one must first analyze each of the potential situations to predict the outcomes and then pick the best strategy. In a two-player game, this would take into account the various strategies available to both players. A Nash equilibrium is reached when neither player alters their strategy after learning all about the other’s strategy.

Nash equilibrium vs. Dominant strategy

Nash equilibrium is frequently compared with the dominant strategy, both of which are game theory strategies. As we saw above, the Nash equilibrium argues that the best strategy for a player is to stick to their initial plan while knowing the opponent’s approach and that all participants maintain the same strategy. On the other hand, the dominant strategy argues that a player’s selected approach will provide superior outcomes than all other potential strategies, despite the strategy employed by the opponent.

The bottom line

Although widely used to make economic and business decisions, the Nash equilibrium in non-cooperative game theory has its own set of limitations too. One of the most important drawbacks of the Nash equilibrium is that it necessitates individuals to be aware of their opponent’s strategy. However, when players know their opponent’s strategy, a Nash equilibrium can only emerge if they stick with their current strategy. Nonetheless, Nash equilibrium can be used in a number of real-life circumstances to determine the optimal payout in a scenario depending on the actions of a player and their opponent.

Shield DAO

An open global decentralized governance organization, Shield DAO is a highly secure and stable derivatives trading protocol that is completely based on a non-cooperative game network. Five roles sustain the decentralized network of the Shield protocol:

  • Brokers
  • Traders
  • Private pool
  • Public pool, and
  • Liquidators

The platform’s security and stability are primarily maintained through attaining an ideal network-wide Nash equilibrium, and each of these roles plays a key part in this. Moreover, each participant that contributes to the network’s upkeep is also incentivized. And it is through this network that Shield has built its perpetual options, a first it's kind long-term on-chain option that allows traders to trade without the effort, risk, or expense of rolling positions.

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