The Concept of Game Theory - Example

THE GAME THEORY Game theory Introduction Most business corporations and companies are turning to the concept of game theory to guide them in making strategic decisions that concern their operations. The concept is the employment of mathematical and analytical designs that guide managers to understand particular situations and thus make decisions. Game theory in general terms can be defined as the process of applying mathematical models of conflict and cooperation to analyze the interaction between two or more players in a situation with set rules and outcomes. In economic the game theory tool is valuable in aiding fundamental analysis of various industry sectors and any premeditated interaction involving two or more firms. However, the concept is not new as it has been employed before in business to help determine pricing of products, labor negotiation decisions and designing marketing strategies. Therefore, the game theory tools, when followed to the later, can be powerful strategic support tools in the decision-making process. To clearly understand the concept of game theory, one needs to know the game, the players, the strategy, and the expected outcome. According to Myerson (1991), in such a situation the game will refer to the circumstances that have a result dependent on actions of two or more decision makers who are players. Players, in this case, will be the strategic decision makers within the frame of the game where the strategy will be the complete plan of action one will take depending on the circumstance that may arise in the game. Making a particular decision, one expects an outcome out of that and in game theory it is referred to payout that can be quantifiable to show any achievements. Once agreements have been arrived at by both players making their decisions, then the game will be set at equilibrium. Therefore, the game is the interaction between these players and available moves that are strategies relevant to specific payoffs for every combination. Simultaneous Move Games with Complete Information In the gaming theory, the most basic game between players is one where they choose their strategies simultaneously thus they make the moves without knowledge of each others strategy. Presumptions are made in the game that there is complete information where each player knows the other’s payoff from each outcome thence the game becomes less complicated. Therefore, simultaneous games are imperfect information games since players can not exactly pinpoint the level of the game as they are unable to tell the opponents next strategy prior to making their decision. An example of a simultaneous move game is rock, paper and scissors in which players make their choice at the same moment independent of each other. Tadelis (2013) notes that, the best way to represent simultaneous-move games is in matrix form where player one selects a row at the same time player two selects a column and thus the merging cell is the outcome of the game. Alternatively the game can be represented by the game tree where information sets are used to emphasize the imperfect information. They provide a possibility of having more than two players although it is also associated with its complexities. Strategies can either be strongly or weakly dominant, and one is considered dominant if it is the best combination of action than the other players that are if it gives higher payoff than all the other strategies. Comparatively a strategy will be strictly dominated if and only if there is another strategy that gives the other player a comparative advantage with higher payoff for all the possible combinations of other players. Essentially it is easier to predict that a player will play their dominant strategy and never a dominated one thus making it possible to predict the outcome To solve the issues of the dominance of strategies, one has to employ iterated strict dominance a means of deleting that strategy that has been dominated with respect to the other remaining strategies. The process is repeated till there are no dominated strategies. Therefore, the game is solvable if the iterated strict dominance leads to an exclusively predicted payoff that is only one player comes out on top. Solving of dominance brings the game into an equilibrium that is a state of balance where neither strategy holds an advantage over the other. The solution in economic terms is what can refer to as Nash equilibrium that is a set of strategies where no individual player can unilaterally deviate from these strategies and hold an advantage of being strictly better off. In this case, both players will be contented thus there will be no need to switch their strategies from what everyone is doing. Nash equilibrium is a pure strategy where each player’s strategy puts all weight o an action. An example of Nash equilibrium is the prison dilemma where two suspects are held in different cells. The available evidence is not enough to convict them of the major crime, but the only minor offense and they can only be convicted of a major crime if they act as informers against each other. If both choose to stay quiet they therefore only serve shorter sentence but if one of the chooses to confess they will be acquitted while the other serve a length sentence. The situation is modeled as a game where the players are the suspects, and the actions are to either confess or defect. The model can be represented in a payoff matrix as shown (Oyama, 2002). C D C 5,5 0,10 D 10,0 1,1 There are four possible combinations of actions in the game, for instance, the action pair (confess, confess) is a pure Nash equilibrium strategy because if the player 2 confesses then player 1 will be better off if the confess rather than defect. In simpler terms in the game, there is no need to change strategy if the other player has not changed theirs. Simultaneous Move Games with Incomplete Information Mostly players will not have complete information about their opponents, for instance, a seller of a car may not have an idea of the potential valuation of the buyer. Such situations are analyzed by converting the game into one of imperfect information where the players do not correctly observe their opponents strategies or forget theirs. The kind of game, where there is imperfect information, is referred as a Bayesian game. The Bayesian principle is common in auctions where for instance if someone wants to sell a house they will auction it to the highest bidder (Harsanyi, 2004). Each buyer will have their private valuation of the house, and they will submit their bid simultaneously. The highest bidder wins the auction and pays their bid and in circumstances of a tie coin will be flipped. In this situation, each player does not know what action their opponent will take since there is incomplete information about their actions. Bayesian Nash equilibrium is, therefore, a straightforward extension of the Nash equilibrium where each player chooses a strategy that maximizes the expected utility given the actions of opponents and the player’s belief about the others’ actions. The action of the first player in the game optimizes the expected utility given the actions of player two. Therefore, payoff for a player in a Bayesian game relies on the strategies of all players in the game and understanding each type of player. In most cases, incomplete information cannot be represented as strategic from games thus there is a need to consider extensive form games with the precise order of moves. In this case, information sets are used to represent what each players capabilities at every stage of the game. Because the games are dynamic there is, therefore, need to strengthen the Bayesian Nash equilibrium so that it has some perfection. Therefore, a perfect Bayesian Nash equilibrium in changing game of imperfect information is a strategy profile and belief system in which that strategy is sequentially given, and the belief system is consistent. Perfect Bayesian Nash equilibrium is a weak equilibrium concept for games with incomplete information that is always strengthened by limited beliefs of the information set that does not lie along the equilibrium path. Sequential Move Games with Complete Information Sequential games in which moves are not simultaneous with imperfect information. These games can be divided according to whether players who make first moves access information that is hidden from those who will make their moves later. Slantchev (2005) affirms that, in sequential games, game trees are mostly used to specify the possible moves of the players and expected outcome and payoffs. Firm 1 Buy Dont Firm 2 Buy don’t buy Buy don’t buy 3,3 4,1 1,4 2,2 Branches of the tree provide the possible choices the players have for instance in the example above they will either buy or dont buy. The nodes that are the root and terminal nodes are places where the players will take action to make a move. In incomplete information, sequential games at least some information is provided to a player that an opponent is moving afterward does not have. Players who have prior information may wish to signal its nature through their actions to opponents who will move later provided they are running under similar objectives. Similarly, an informed play may disguise their knowledge by taking actions that do not reveal their nature. To solve the game tree, backward induction or rollback is employed to bring the game to equilibrium. However, the process may not predict behavior if players are overly concerned with just behavior by their opponents and that they do not take action to optimize their payoffs. Sequential Move Games with Incomplete Information In sequential move games with incomplete information, the perfect Nash equilibrium apply in the sequential bargaining model similar to complete information games. Compared to situations where there is complete information where Nash equilibrium is based on assumptions that players will act sequentially irrational for the case of incomplete information the presumption is that each agent’s action is sequentially rational (Fudenberg & tirole, 1991). A situation where there is sequential rationality in game theory is what is referred as perfect Bayesian Nash equilibrium. For each information set, beliefs of the agent are specified, and they are represented by a probability distribution on the information set. Players demonstrate sequential rationality if at each information set they move they optimize the expected utility given their beliefs at the information set even if the information is precluded by their strategy. If player 1 has a dominant strategy, then his move is sequentially rational although there is a challenge in that player 2s beliefs might not be consistent with player 1s strategy. Comparatively I equilibrium the player will maximize the expected payoff considering the opponent’s strategy. For mixed strategies, the information set is said to be on the path of play provided the information set is reached upon with positive probability. Signaling is a sequential dynamic game of incomplete information that involves only two players that are the sender and receiver. It is a straightforward model that allows the information sets dependent on the nature to draws actions that determine what the player will choose from. In job market signaling, the sender of information is the worker, and the receiver is the market of potential employees. The productive ability of the worker represents the type will education choice of the worker is the message that brings the action in the form of the wage paid in the market. Sometimes the signaling as a game is embedded in the whole game where, for instance, there could be no action by the receiver before the sender chooses the strategy. Payoff to the worker is the cost of the workers ability in obtaining an education. In a situation of perfect Bayesian equilibrium, the firm will interpret education as a signal of strength thus they will offer more wages to a worker with more education. In incomplete information game, the cheap talk will refer to the interaction between players that will ultimately not affect the games payoffs. According to de Berreda (2010), cheap talk is in contrast to signaling where information sending is costly for the player that is sending the message, depending on the state of nature. For instance, if an expert is trying to explain the state of the world to a layman who wants to invest in business who will eventually make a decision that will affect both payoffs. Therefore, cheap talk can be factored in anywhere since it has the potential to enhance the outcomes in an equilibrium state. In situations where its only one player is messaging and the other receives then the first player holds a first mover advantage over the opponent. However, it is not guaranteed that cheap talk will influence the equilibrium payoff. Monopoly screening refers to a situation where the seller screens different types of buyers that is they induce them to select varied consumption bundles. It is at times referred as second-degree price determination where the monopolist cant identify the consumers correctly. For instance on a given day people will have the reason to act like low valued users thus pay lower prices. In order for the seller to be in a position to charge high-valued users highly then they have to design a pricing scheme in which high valued individual will not act like low valued people. Thus, the whole process of coming up with a pricing scheme in a monopoly is what is called screening. The monopolist chooses their strategy by changing the bundles since they offer options more so if the bundles for the high valued customers and those for the lowly valued consumers are offered at the same superfluous to the highly valued consumers then they will only choose the bundle intended for them. Similarly, if the monopolist decreased the prices charged to the higher bundle, then they would earn surplus from choosing the bundle and hence they will go for it. Example Real Life Application of Game Theory In real life game theory can be applied where changing significant action brings about changes in the outcome. For example consider a situation where Bath city council provides wastes recycling centers to recycle as much waste as possible. In normal circumstances anyone could use the recycling centers whether they are residents of the city or non residents. Similarly Bath city residents can use the recycling services of another city apart from their own. Assume that the council spends around $100,000 on the recycling service and that 10%of the users are non residents who if banned will see the cost decrease to $90,000 but the council then has to pay monitoring cost of who uses the sites. Therefore if the cost of monitoring is $5,000 then the total cost of the council becomes $95,000 and thus the council could have saved $5,000 by banning non residents. In contrast the other city will have more people to deal with since their residents cannot use the services of Bath city hence their cost will increase to $110,000. The move is fine for Bath city till their neighbor decides to take the same action of banning non residents from using their recycling services. In such a situation the 10% who have been using the neighboring centre can only use the centers in Bath city and hence the $10,000 will add up the cost to $105,000 which is the cost of the service including monitoring users. The game can be represented in a matrix as Without ban of non residents Ban non resident Without ban of non residents $100,000; $100,000 $110,000; $95,000 Ban non residents $95,000; $110,000 $105,000; $105,000 The game presents a classic prisoners’ dilemma in that if the council of the Bath city bans non residents then its cost will be $95,000 rather than $100,000 that provided the other city does not take any action. Alternatively the council of the neighboring city may also ban non residents in response to the Bath city’s move and thus their cost will be $105,000 rather than $110,000 when they don’t take any action. Either way the best response for both cities is to restrict their services although it leads to a worse result compared to if they kept an open system. Although these services have been running over time without restriction the change of game has results from the changes in economic times. Prior to changes the councils’ service was based on maximizing the waste recycled thus there was no need of restriction but now the councils are playing the game using money as a measure of success hence the councils are trapped in a prisoners’ dilemma in terms of game. References de Barreda, I. M. (2010). Cheap Talk With Two-Sided Private Information Job Market Paper. Fudenberg, D., & Tirole, J. (1991). Perfect Bayesian equilibrium and sequential equilibrium. journal of Economic Theory, 53(2), 236-260. Harsanyi, J. C. (2004). Games with Incomplete Information Played by “Bayesian” Players, I–III: Part I. The Basic Model&. Management science, 50(12_supplement), 1804-1817. MYERSON, R. B. (1991). Game theory analysis of conflict. Cambridge, Mass, Harvard University Press Oyama, D. (2002). p-Dominance and equilibrium selection under perfect foresight dynamics. Journal of Economic Theory, 107(2), 288-310 Slantchev, B. L. (2005). Game Theory: Dominance, Nash Equilibrium, Symmetry. TADELIS, S. (2013). Game theory an introduction. Princeton, NJ, Princeton University Press Read More