1. Game Theory Optimal Gto Strategy
2. Game Theory Optimal Poker Strategy
3. Game Theory Examples In Business
4. Optimal Strategy Game Theory Matrix

In 1912 the German mathematician Ernst Zermelo proved that such games are strictly determined; by making use of all available information, the players can deduce strategies that are optimal, which makes the outcome preordained (strictly determined). Game theory is a way of thinking about strategic situations. One aim of the course is to teach you some strategic considerations to take into account making your choices. A second aim is to predict how other people or organizations behave when they are in strategic settings.

  Strategy profile sis Pareto optimal, or strictly Pareto efficient, if there’s no strategy s' that Pareto dominates s   Every game has at least one Pareto optimal profile   Always at least one Pareto optimal profile in which the strategies are pure Nau: Game Theory 9. Well, obviously, throwing one third, one third, one third would be the game theory optimal approach. The problem with that is while you won't lose, you also won't win if you stick to that- unless you get lucky, of course. But that's not an exploitative strategy. Now, what if you find- you notice that your opponent throws rock a lot?

• Two-person constant-sum games
• Two-person variable-sum games
  • The prisoner’s dilemma
• N-person games

Games of perfect information

The simplest game of any real theoretical interest is a two-person constant-sum game of perfect information. Examples of such games include chess, checkers, and the Japanese game of go. In 1912 the German mathematician Ernst Zermelo proved that such games are strictly determined; by making use of all available information, the players can deduce strategies that are optimal, which makes the outcome preordained (strictly determined). In chess, for example, exactly one of three outcomes must occur if the players make optimal choices: (1) White wins (has a strategy that wins against any strategy of Black); (2) Black wins; or (3) White and Black draw. In principle, a sufficiently powerful supercomputer could determine which of the three outcomes will occur. However, considering that there are some 10⁴³ distinct 40-move games of chess possible, there seems no possibility that such a computer will be developed now or in the foreseeable future. Therefore, while chess is of only minor interest in game theory, it is likely to remain a game of enduring intellectual interest.

Games of imperfect information

A “saddlepoint” in a two-person constant-sum game is the outcome that rational players would choose. (Its name derives from its being the minimum of a row that is also the maximum of a column in a payoff matrix—to be illustrated shortly—which corresponds to the shape of a saddle.) A saddlepoint always exists in games of perfect information but may or may not exist in games of imperfect information. By choosing a strategy associated with this outcome, each player obtains an amount at least equal to his payoff at that outcome, no matter what the other player does. This payoff is called the value of the game; as in perfect-information games, it is preordained by the players’ choices of strategies associated with the saddlepoint, making such games strictly determined.

The normal-form game in Table 1 is used to illustrate the calculation of a saddlepoint. Two political parties, A and B, must each decide how to handle a controversial issue in a certain election. Each party can either support the issue, oppose it, or evade it by being ambiguous. The decisions by A and B on this issue determine the percentage of the vote that each party receives. The entries in the payoff matrix represent party A’s percentage of the vote (the remaining percentage goes to B). When, for example, A supports the issue and B evades it, A gets 80 percent and B 20 percent of the vote.

Assume that each party wants to maximize its vote. A’s decision seems difficult at first because it depends on B’s choice of strategy. A does best to support if B evades, oppose if B supports, and evade if B opposes. A must therefore consider B’s decision before making its own. Note that no matter what A does, B obtains the largest percentage of the vote (smallest percentage for A) by opposing the issue rather than supporting it or evading it. Once A recognizes this, its strategy obviously should be to evade, settling for 30 percent of the vote. Thus, a 30 to 70 percent division of the vote, to A and B respectively, is the game’s saddlepoint.

A more systematic way of finding a saddlepoint is to determine the so-called maximin and minimax values. A first determines the minimum percentage of votes it can obtain for each of its strategies; it then finds the maximum of these three minimum values, giving the maximin. The minimum percentages A will get if it supports, opposes, or evades are, respectively, 20, 25, and 30. The largest of these, 30, is the maximin value. Similarly, for each strategy B chooses, it determines the maximum percentage of votes A will win (and thus the minimum that it can win). In this case, if B supports, opposes, or evades, the maximum A will get is 80, 30, and 80, respectively. B will obtain its largest percentage by minimizing A’s maximum percent of the vote, giving the minimax. The smallest of A’s maximum values is 30, so 30 is B’s minimax value. Because both the minimax and the maximin values coincide, 30 is a saddlepoint. The two parties might as well announce their strategies in advance, because the other party cannot gain from this knowledge.

Mixed strategies and the minimax theorem

When saddlepoints exist, the optimal strategies and outcomes can be easily determined, as was just illustrated. However, when there is no saddlepoint the calculation is more elaborate, as illustrated in Table 2.

A guard is hired to protect two safes in separate locations: S1 contains $10,000 and S2 contains $100,000. The guard can protect only one safe at a time from a safecracker. The safecracker and the guard must decide in advance, without knowing what the other party will do, which safe to try to rob and which safe to protect. When they go to the same safe, the safecracker gets nothing; when they go to different safes, the safecracker gets the contents of the unprotected safe.

In such a game, game theory does not indicate that any one particular strategy is best. Instead, it prescribes that a strategy be chosen in accordance with a probability distribution, which in this simple example is quite easy to calculate. In larger and more complex games, finding this strategy involves solving a problem in linear programming, which can be considerably more difficult.

To calculate the appropriate probability distribution in this example, each player adopts a strategy that makes him indifferent to what his opponent does. Assume that the guard protects S1 with probability p and S2 with probability 1 − p. Thus, if the safecracker tries S1, he will be successful whenever the guard protects S2. In other words, he will get $10,000 with probability 1 − p and $0 with probability p for an average gain of $10,000(1 − p). Similarly, if the safecracker tries S2, he will get $100,000 with probability p and $0 with probability 1 − p for an average gain of $100,000p.

The guard will be indifferent to which safe the safecracker chooses if the average amount stolen is the same in both cases—that is, if $10,000(1 − p) = $100,000p. Solving for p gives p = 1/11. If the guard protects S1 with probability 1/11 and S2 with probability 10/11, he will lose, on average, no more than about $9,091 whatever the safecracker does.

Using the same kind of argument, it can be shown that the safecracker will get an average of at least $9,091 if he tries to steal from S1 with probability 10/11 and from S2 with probability 1/11. This solution in terms of mixed strategies, which are assumed to be chosen at random with the indicated probabilities, is analogous to the solution of the game with a saddlepoint (in which a pure, or single best, strategy exists for each player).

The safecracker and the guard give away nothing if they announce the probabilities with which they will randomly choose their respective strategies. On the other hand, if they make themselves predictable by exhibiting any kind of pattern in their choices, this information can be exploited by the other player.

The minimax theorem, which von Neumann proved in 1928, states that every finite, two-person constant-sum game has a solution in pure or mixed strategies. Specifically, it says that for every such game between players A and B, there is a value v and strategies for A and B such that, if A adopts its optimal (maximin) strategy, the outcome will be at least as favourable to A as v; if B adopts its optimal (minimax) strategy, the outcome will be no more favourable to A than v. Thus, A and B have both the incentive and the ability to enforce an outcome that gives an (expected) payoff of v.

Utility theory

In the previous example it was tacitly assumed that the players were maximizing their average profits, but in practice players may consider other factors. For example, few people would risk a sure gain of $1,000,000 for an even chance of winning either $3,000,000 or $0, even though the expected (average) gain from this bet is $1,500,000. In fact, many decisions that people make, such as buying insurance policies, playing lotteries, and gambling at a casino, indicate that they are not maximizing their average profits. Game theory does not attempt to state what a player’s goal should be; instead, it shows how a player can best achieve his goal, whatever that goal is.

Von Neumann and Morgenstern understood this distinction; to accommodate all players, whatever their goals, they constructed a theory of utility. They began by listing certain axioms that they thought all rational decision makers would follow (for example, if a person likes tea better than coffee, and coffee better than milk, then that person should like tea better than milk). They then proved that it was possible to define a utility function for such decision makers that would reflect their preferences. In essence, a utility function assigns a number to each player’s alternatives to convey their relative attractiveness. Maximizing someone’s expected utility automatically determines a player’s most preferred option. In recent years, however, some doubt has been raised about whether people actually behave in accordance with these axioms, and alternative axioms have been proposed.

When we hear the term “game,” we usually start thinking about amusements or sports. But in a branch of mathematics called “Game Theory,” the word ‘game’ has a much broader connotation. Game Theory is, in fact, the study of mathematical models and their interaction with the decision-makers. The game theory includes strategic thinking in which players make decisions by viewing various perspectives and by looking at the viewpoint of other participant players; also by analyzing their actions and reactions in particular situations.

Significance

The best use of game theory is to figure out the optimal solution from the best possible choices through the analysis of costs and benefits to each participant who competes with each other. The theory is applicable in different fields, like in business, psychology, biology, economics, political science, computers, etc. The practical implication of the theory lies in its use to support in explaining events and situations that occurred in the past and to determine different actions players may take in the future, etc. From the business point of view, game theory can be utilized by business managers to predict the strategic planning or thought process of their competitors and collaborators. It is considered a very powerful medium for forecasting or predicting the output of interactions between different participants or competitors in which reaction of one depends on the action of others.

Components or Elements of Game Theory

The game theory has mainly three components:

• Players: Game theory’s main element includes a set of players. Business managers and companies are considered as players.

• Strategies: Different strategies that players can use in the game. This also includes game rules that are set in a hierarchy to mention the arrangement of the best possible strategies and actions.

• Outcomes or results: This includes the possible strategies and expected payoffs as per outcomes. It is assumed that all the participants are already aware of these payoffs.

Game Theory Optimal Gto Strategy

There are two main ways by which the game theory can be used, i.e., simultaneous games in which the players go ahead with their moves or actions simultaneously, irrespective of looking for information related to the moves or actions chosen by other players. The other one is a sequential game, which includes the dependency of the player’s moves or actions on the previous action’s results or another player’s choice.

Examples of Game Theory

Let’s have some real-life examples of Game Theory.

1. Bidding at Auction

An auction is considered as a sale activity in which different bidders bid for purchasing any good or service, and the same is sold to the bidder having the highest bid. Game theory is being used in bid auctions, especially for analysis of the first price sealed auction bidding. In this type of auction, bidders are supposed to submit bids in safe and sealed way. Different players are unaware of the value of goods or services of each other and try to devise a bidding strategy, irrespective of having complete information. The behavior of bidders and different other factors are being analyzed while making decisions during bid preparation.

2. Collective Bargaining or Negotiation Between Parties

Game theory plays an important role in different collective bargaining or negotiation activities among different parties or participants. For example, different negotiations take place between worker unions and the management during the situation of the strike of workers or lockout period to increase wages. Using game theory, both parties can arrive at the optimum solution of the issue, i.e., to increase wages by examining different options available for wages and benefits, which can maximize the welfare of both workers and management. Salary negotiation is also an example of the game theory application. The concept of game theory is used in other negotiations also like negotiations with suppliers while purchasing, compensation or incentive negotiations between management and suppliers or business partners, etc.

Game Theory Optimal Poker Strategy

3. Decisions Related to New Products

Product-based decisions related to launching a new product in the market or to exit the launch of the product are also executed using game theory by businesses. Using game theory, business people can understand if the first-mover advantage is there or not, competitor’s possible moves related to new products and also deciding upon the strategies for defense, etc. Similarly, the concept of game theory is also being used in deciding whether entering a new market or exiting it.

4. Product Pricing Decisions

Game Theory Examples In Business

The applications of game theory are also being used widely in deciding upon the pricing strategies of both consumers and retailers. Retailers compete against each other to gain market share of customers and for this, they opt for different games or strategies, like offering attractive discounts on specific goods in order to increase sales of complementary products. For example, in the offseason, i.e., off-summer or off-winter season, garment shop vendors or retailers offer attractive sales on a certain stock of clothes in which they adopt optimal pricing strategies to attract maximum customers. In this, retailers use the game theory approach where retailers and consumers are the main players. The focus of retailers is on using the best pricing strategy while the preference of consumers is to choose the best deal in terms of discount and variety.

5. Stock Market Decisions

Optimal Strategy Game Theory Matrix

Using game theory, decisions regarding buying and selling shares in the stock market can be taken wisely. The investors make different stock market decisions by using different strategies of investment and by considering different players or investors. Game theory helps in predicting the decisions of other players related to investments, and based on these decisions, they can decide upon the strategies for themselves which maximize profit.