Game theory and application in economics

Nash proved that every finite n-player, non-zero-sum not just two-player zero-sum non-cooperative game has what is now known as a Nash equilibrium in mixed strategies.. Players: Individu

FOREIGN TRADE UNIVERSITY VIETNAM - JAPAN HUMAN RESOURCES DEVELOPMENT

INSTITUTE (VJCC)

GAME THEORY AND APPLICATION IN ECONOMICS

Student’s full name : Le Hien Trang Student ID : 2113520021 Class : A02 - High-quality program Intake : 60

Ha Noi, December 2022

TABLE OF CONTENT

A THEORETICAL BASIS

I Definition

II Historical overview

1 Precursors

2 Birth and early development

III Important Terms

IV Categorization

1 Game performance

2 Types of Game theory

a) Cooperative & Non-cooperative games

b) Complete-information & Incomplete-information games

c) Symmetric & Asymmetric games

d) Constant-sum, Zero-sum & Non-zero-sum games

e) Simultaneous-move & Sequential-move games

f) One-Shot & Repeated games

B PRACTICAL BASIS

I Purposes of Game theory in economics

II Application of Game theory in economics

A THEORETICAL BASIS Definition

Game theory is a branch of applied mathematics that provides tools for analyzing situations in which parties, called players, make decisions when they are interdependent This interdependence causes each player to consider the other player’s possible decisions, or strategies, in formulating strategy

Historical overview

1 Precursors

Discussions on the mathematics of games began long before the rise of modern mathematical game theory Cardano's work on games of chance in “Book on Games of Chance’, written around 1564 but published posthumously in 1663, formulated some of the field's basic ideas

In 1713, a letter attributed to Charles Waldegrave analyzed a game called "le Her" In this letter, Waldegrave provided a minimax mixed strategy solution to

a two-person version of the card game le Her In his 1838 “Researches into the Mathematical Principles of the Theory of Wealth”, Antoine Augustin Cournot considered a duopoly and presented a solution that is the Nash equilibrium of the game

In 1913, Ernst Zermelo published “On an Application of Set Theory to the Theory of the Game of Chess”, which proved that the optimal chess strategy is strictly determined This paved the way for more general theorems

In 1938, the Danish mathematical economist Frederik Zeuthen proved that the mathematical model had a winning strategy by using Brouwer's fixed point theorem In his 1938 book Applications aux Jeux de Hasard and earlier notes, Emile Borel proved a minimax theorem for two-person zero-sum matrix games only when the pay-off matrix is symmetric and provided a solution to a non-trivial infinite game (known in English as Blotto game) Borel conjectured the non-existence of mixed-strategy equilibria in finite two-person zero-sum games, a conjecture that was proved false by von Neumann

Il

2 Birth and early development

Game theory did not exist as a unique field until John von Neumann published the paper On the Theory of Games of Strategy in 1928 His 's original proof used Brouwer's fixed-point theorem on continuous mappings into compact

mathematical economics His paper was followed by his 1944 book Theory of Games and Economic Behavior co-authored with Oskar Morgenstern His paper was followed by his 1944 book Theory of Games and Economic Behavior co-authored with Oskar Morgenstern

In 1950, the first mathematical discussion of the prisoner's dilemma appeared, and an experiment was undertaken by notable mathematicians Merrill M Flood and Melvin Dresher, as part of the RAND Corporation's investigations into game theory RAND pursued the studies because of possible applications to global nuclear strategy Around this same time, John Nash developed a criterion for mutual consistency of players’ strategies known as the Nash equilibrium, applicable to a wider variety of games than the criterion proposed

by von Neumann and Morgenstern Nash proved that every finite n-player, non-zero-sum (not just two-player zero-sum) non-cooperative game has what is now known as a Nash equilibrium in mixed strategies

Game theory experienced a flurry of activity in the 1950s, during which the concepts of the core, the extensive form game, fictitious play, repeated games, and the Shapley value were developed The 1950s also saw the first applications of game theory to philosophy and political science

In the 1970s, game theory was extensively applied in biology, largely as a result of the work of John Maynard Smith and his evolutionarily stable strategy In addition, the concepts of correlated equilibrium, trembling hand perfection, and common knowledge were introduced and analyzed

Important Terms

Game: Any situations that have results depending on the actions of two or more decision-makers (players)

IV

Players: Individuals who make decisions

Strategy: a decision rule that describes the actions a player will take at each decision point

Information: Information available at one identified moment of the game

decision-making and results are formed accordingly

Payoff: An amount showing as an element in the payoff matrix, which indicates the amount gained or lost by the row player

Payoff matrix: A matrix whose elements represent all the amounts won or lost

by the row player

Categorization

1 Game performance

There are two forms that the game could be displayed: Normal-form game and Extensive-form game

a) Normal-form game

PLAYER 2

Strategy A Pi Ps, P,,* Py

PLAYER 1

The normal form (or strategic form) is a way of describing a game using a matrix The normal-form representation of a game includes all perceptible and conceivable strategies (here strategy A and strategy B), and their corresponding payoffs, for each player (here p1A,p2A; p1A,p2B; p1B,p2A; p1B,p2B) The strategic form allows us to quickly analyze each possible outcome of a game In the depicted matrix, if player 1 chooses strategy A and player 2 chooses strategy B, the set of payoffs given by the outcome would be plA,p2B

If player 1 chooses strategy B and player 2 chooses strategy A, the set of payoffs would be p1B,p2A

The strategic form is usually the right description for simultaneous games, where both players choose simultaneously

b) Extensive-form game

Piy Pr,

Pay Pạ; PLAYER 1

Pig Pa

Pia Po

The extensive form is a way of describing a game using a game tree This form

is a specification of a game in game theory, allowing for the explicit representation of many key aspects, like the sequencing of players' possible moves, their choices at every decision point, the (possibly imperfect) information each player has about the other player's moves when they make a decision, and their payoffs for all possible game outcomes The payoffs are represented at the end of each branch

In the depicted tree, if player 1 chooses strategy A and player 2 chooses strategy B, the set of payoffs will be p1A,p2B

Since the extensive form represents decisions at different moments, it is usually used to describe sequential games

2 Types of Game theory

a) Cooperative & Non-cooperative games

(1) | Cooperative

Cooperative game models how agents compete and cooperate as coalitions in

coalitions of players rather than between individuals, and it questions how groups form and how they allocate the payoff among players

contact each other, they must have decided to remain silent Therefore, their

negotiation would have helped in solving the problem

(2) | Non-cooperative

Non-cooperative game models the actions of agents, maximizing their utility in

a defined procedure, relying on a detailed description of the moves and information available to each agent

The most common non-cooperative game is the strategic game, in which only the available strategies and the outcomes that result from a combination of choices are listed Rock-paper-scissors is one of many examples of this type of game

b) Complete-information & Incomplete-information games

Complete information is the concept that each player in the game is aware of the sequence, strategies, and payoffs throughout gameplay Given this information, the players can plan accordingly based on the information to maximize their strategies and utility at the end of the game

Incomplete Information game is a game where the players do not have common knowledge of the game being played This idea is tremendously important in capturing many economic situations, where a variety of features of the environment may not be commonly known

c) Symmetric & Asymmetric games

In this game, strategies adopted by all players are the same Each player earns the same payoff when making the same choice against similar choices of his competitors

Symmetry can exist in short-term games only because in long-term games the number of options with a player increases The decisions in a symmetric game

depend on the strategies used, not on the players of the game Even in the case

of interchanging players, the decisions remain the same Many 2x2 games are considered Symmetric ones

General form

Player 2

Player |

Player 2’s payoff

a Cc

a Cc

Player 1’s Payoff

In this game, strategies adopted by players are different The strategy that provides benefit to one player may not be equally beneficial for the other player However, decision-making in asymmetric games depends on the different types of strategies and decisions of players

d) Constant-sum, Zero-sum & Non-zero-sum games

Constant-sum game is one in which the sum of outcomes of all the players remains constant even if the outcomes are different

Poker, for example, is a constant-sum game because the combined wealth of the players remains constant, though its distribution shifts in the course of play

Zero-sum game is a type of constant sum game in which the sum of outcomes

of all players is zero The collective net benefit received is equal to the collective net benefit lost

Almost every sporting event is a zero-sum game in which one team wins and

one team loses

Player 2

Player I

(3) Non-zero-sum

A non-zero-sum game is one in which all participants can win or lose at the same time A non-zero-sum game can be transformed into the zero-sum game

by adding one dummy player The losses of dummy players are overridden by the net earnings of players

For example, in gambling or chess, the gain of one player results in the loss of the other player

e) Simultaneous-move & Sequential-move games

Simultaneous games are the ones in which the move of two players is simultaneous This means each participant must continually make decisions at the same time their opponent is making decisions In a simultaneous move, players do not have knowledge about the move of other players

Simultaneous games are represented in normal form

For instance, as companies devise their marketing, product development, and operational plans, competing companies are also doing the same thing at the same time

Sequential games are the ones in which players are aware of the moves of players who have already adopted a strategy However, the players do not have deep knowledge of the strategies of other players

Sequential games are represented in extensive form

For example, a player has knowledge that the other player would not use a single strategy, but he/she is not sure about the number of strategies the other player may use

f) One-Shot & Repeated games

One-shot game is a type of game theory that is played once only In one-shot interaction, people often have the incentive to behave opportunistically

This is often the case with equity traders that must wisely choose their entry point and exit point as their decision may not easily be undone or retried

Repeated game is played more than once; either with a finite or infinite number

of interactions It is a very common feature of applied business such as pricing, capital investment, supply capacity forecasts, and research and development spending, to name but a few With repeated games, there is more scope for cooperative strategies to emerge

Take rival companies trying to price their goods into consideration, whenever one makes a price adjustment, so may the other This circular competition repeats itself across product cycles or sale seasonality

B PRACTICAL BASIS

I Purposes of Game theory in economics

therefore, firm leaders are able to make optimal decisions more easily To be particular, the game theory would help players identify appropriate ways to deal with or cooperate with opponents

humans in socio-economic life from their perspective of individual and collective strategies

to one individual might become disadvantageous to a whole group

II Application of Game theory in economics

auction, bargain, and market types including duopoly, oligopoly, to name but a few

Back to the time when Coca Cola just emerged into Vietnamese market, its strategy to steal the market share was considerably lowering product price Understanding this situation, Pepsi immediately utilized the same stategy as Coca Cola to compete The game theory application of these two firms would

be illustrated in the table below: