The absence of a dominant strategy allows for the existence of a point where each player takes up a strategy that maximizes his or her outcome given the other player's options. This is what is known as the Nash equilibrium. At this point, the players take up their own individual choices and cannot improve their outcome given that the prevailing circumstances. Satisfaction of the Nash equilibrium follows from rational choices by players and the expectation that opponent players will follow the same route. [15: Rasmusen, & Eric. (2001). Games and Information: An Introduction to Game Theory, 3rd ed. . Oxford: Blackwell.]
Considering the internet service example given above, the interactions with the game can be altered by inclusion of a clause where the consumer can opt-out of the service contract. This clause safeguards the interest for the consumer to receive a high quality service as the producer is given an incentive to do so. The resultant game interaction is such that the consumer will take up high quality service since it has the highest payoff. Similarly, the service provider will take up provision of high quality service since it is in their best interests give the opt-out clause included.
The resulting interactions yield two equilibrium position. The first position is where both player maximize their gain by not buying (consumer) and provide low quality service (service provider). The other equilibrium is the Nash equilibrium where preference is to buy and provide high quality service. This equilibrium provides a high pay to the players in the game and therefore, it is a preferred option. It is in the best interest for the players to make no alterations on their choice. In the Nash equilibrium, the player seeks to maximize their gain and is the ideal situation in utility maximization. The Nash equilibrium is attained from the elimination of the dominant strategy that adds to the choices a unique strategy combination. The unique strategy combination is thus the Nash equilibrium.
Selection of equilibrium
In game theory, situations arise where there is more than one equilibrium. This situation calls for a guide to facilitate players through the interactions and help them to pick the most reasonable equilibrium. This will be the position whose utility is highest and meets the preference of the players. Game theory works to assess the equilibrium positions that are convincing and more plausible. Recent attempts of game theorist try to refine the equilibrium and heighten them as plausible and of high utility[footnoteRef:16]. [16: Harsanyi, J.C., & Selten, R. (1987). A General Theory of Equilibrium Selection in Games. Cambridge, Mass: MIT Press.]
Consider two firms that provide communication services to consumers as well as interact between themselves. If these firms have the choice between two strategies high and low bandwidth equipment, their best option for will be derived from the ability to interpret the other player's preferences. The situations the firms face indicate that, low bandwidth connection works equally well similar to high bandwidth connection. So if firm one has low bandwidth. Switching to high bandwidth is only preferable if firm two has high bandwidth. It is notable that switching to high bandwidth while the other firm has low bandwidth will mean incurring unnecessary costs.
In this type of situation, the option for low bandwidth has an inferior payoff compared to the payoff derived from high bandwidth. However high bandwidth has a higher payoff despite the high cost that to some extent is deemed unnecessary. Low bandwidth options is seen to avail a better worst-case gain in the presence or absence of rationality considering all other options available to the other player, in this it is seen that the player will take up low bandwidth option as one that maximizes the minimum attainable payoff. This option makes investment in low bandwidth a safer choice. This is so with the expectation that the other player will apply similar reasoning as opposed to going for high bandwidth. The realized equilibrium is one the will maximize the minimum attainable payoff.
The more practical application of game theory follows from the appreciation that in any economic, political or social scenario there are more than two players. This complicates the rational decision-making process since each of the players in the game can easily opt for any of the strategies given their respective preference and measures of utility. The game dynamic is given by a display of the fact that a specific ratio of the players opts for each of the strategy. Subsequent interplays though increase in the number of players or change in decision by existing players will be dictated by the measures of the better average gains. This will evaluation and changes will eventually contribute to elimination of some strategies and shift to a long-term surviving strategy. This shift yields the long-term equilibrium[footnoteRef:17]. [17: Rasmusen, & Eric. (2001). Games and Information: An Introduction to Game Theory, 3rd ed. . Oxford: Blackwell.]
Initially a smaller proportion of the population, embrace high quality bandwidth compared to those with low quality bandwidth. This is the case owing the simplistic assessment and preference of maximizing a minimum by the players. Combination of the proportion of individuals with high quality bandwidth with the payoff yield a higher average payoff compared to the low quality bandwidth. New entrants in to the game will look at the average payoff of the different strategies and opt for the strategies with the higher average[footnoteRef:18]. [18: Ibid ]
Existing players in the game will also make similar calculations whet they intent to replace their equipment. Subsequent interaction in the game will lead to a shift to a new equilibrium considering the average advantage. This makes the high quality bandwidth strategy a preferred strategy in future compared to the low quality strategy. However, a shift to the high payoff strategy in the future is pegged upon there being a large enough fraction at the start. If the initial proportion of users in not large enough to attract new entrants, the long-term option will remain as the low quality bandwidth. This interaction declares that long-term sustainability of a strategy is dependent upon the initial combination of the particular option. Those strategies with high initial average will remain the more preferred strategy[footnoteRef:19]. [19: Myerson, & Roger, B. (1995). Game Theory: Analysis of Conflict. Cambridge Massachusetts: Harvard University Press.]
Compliance inspection and certification
Game theory finds application in the now evolving field of compliances and licenses of software packages. Considering the benefits likely to result to the consumer purchase and use of the software, the buyer has an incentive to violate the restrictions relating to use of the product. On the side of the vendor, the vendor needs to ascertain that the consumer abides with the regulations.
In this type of gain, Nash equilibrium is where both player chose a strategy that is preferred and maximizes the gains is not feasible[footnoteRef:20]. Considering that undertaking inspection is a hustle for the vender and failure to inspect leads to losses, then the vender has no maxi min option. Alternatively, the buyer has higher yield benefits if they cheat and an avenue to change their strategy given the option taken by the vender. [20: Ibid ]
In this form of game, probability randomized inspection is advised for the case of the vendor. This measure works to reduce the consumer's certainty of the vendor's possible action. This results to the mixed equilibrium strategy. As a measure to come up with maximum benefit, to the vendor as well as reduce the costs incurred if inspection would be carried out always. In this perspective, Game theory is seen to give an option where a challenge exists in determining the preferred action.
This is the case where the players face the challenge of making a decision completely unaware of the strategic choices the by the opponent. This type of game has a constant sum and the players strategically may choose to randomize their action as this yield better chance for higher payoffs. Ideal practical examples of a zero sum game include "rock-paper scissors," Poker chess and checkers[footnoteRef:21]. Zero-sum game can be applied in modeling the computer science concept strategy of "demonic" non-determinism. This rides on the assumption that worst possible outcome is likely to occur when ordering of events is not specific. One can take nature as an opponent antagonistic enough to randomly yield to favorable and unfavorable possible outcome. This allows an individual to conceptualize the worst case likely to result and use it as a benchmark for alternate decisions[footnoteRef:22]. [21: Myerson, & Roger, B. (1995). Game Theory: Analysis of Conflict. Cambridge Massachusetts: Harvard University Press.] [22: Ibid ]
Similar to the Zero-sum game is the randomization in the algorithms theory by Rao. This theorem defines the power of random algorithms. Zero-sum games and randomized algorithms facilitate analysis of online problems. In this case, algorithm receive input on data at a time and give decision based on the input given. This facilitates…