PRACTICAL ALGORITHMS : GAME THEORY, PRISONER DILEMMA OPTIMAL STOPPING

Our brain is exposed to over 400 billion bits of information per second. All Information
start with input coming from different sensory organs, they transform physical inducements such as to heat, sound waves, touch, or photons of light into electrochemical signals. Notably, only 1 bit in 200 thousand bits of information is used and makes us aware of our surroundings. No matter how intelligent humans are, help from computers and artificial intelligence is exceedingly needed. After decades of research and experiments, a new era has emerged where data analysis and algorithm designs rule every aspect of our lives. We rely on mathematical models to dictate the way we conduct business or make a decision. The models influence us in many ways, with a numerical expression often controlling or driving both major and minor choices – which movie to watch or whether to pursue an investment of a lifetime. Engineers and computer scientists have exploited this advantage to the maximum. Google proved in the past that something as minuscule as a shade of color on a web page could have big consequences, 200 million dollars a year big. A Jillian D'Onfro, a journalist from Business Insider, reported in May of 2016 that "In the early days, Google tested over 40 different shades of blue for its links, and the winning hue helped it reel in an extra $200 million a year in ad revenue". Most people likely never even noticed the difference, but revenue and numbers tend to have the edge over a mere opinion or disbelief. In many ways, algorithmic models illuminate the working of the mind and can be applied to solve dilemmas and help strategize our decisions. Some of the most dominant models such as game theory and optimal stopping are in many forms the core of these algorithms. Game theory mostly deals with evaluating decisions and outcomes for a complicated situation. Especially after the emergence of the internet, computer scientists developed algorithms to predict markets, analyze complex systems and develop innovative ideas. Game theory alone landed John Nash a mathematician from West Virginia a Nobel price back in 1994. Optimal stopping, on the other hand, bargains with choosing the best time to deal with a particular option to either maximize results or lessen the cost, thereby calculating the best chances of reaching a favorable outcome. People may combine the two and benefit their matter significantly by exploiting them in their daily decisions such as finding an apartment, a girlfriend, or even a job. Governments use the exact models to deal with issues related to their economy, environment and elections. As such, one may wonder whether a private individual could substantially benefit from such algorithms as well? Should an individual invest the time to learn more about such seemingly powerful systems? Or is it only reserved for big companies and big governments?

Game theory deals with predicting the outcome of games of strategy in which the participants don't have all the information they need about the other participants.
Game theory analysis has direct relevance to the study of the conduct and behavior of firms in markets. The decisions that firms must take over pricing production and development. However, individual can apply the same strategy to achieve the desired outcome just like a firm would. First, to understand this theory properly this classic example illustrates the core of game theory "The Prisoner's Dilemma". Two prisoners are separated into two different rooms and interrogated over their guilt or innocence of a crime. They both have a choice to make, either to confess to the crime and accept a sentence of 5 years or to deny any criminal activities and hope that their partner does the same hence both walk out free. Now the dilemma lies in the alternative situations. If one of the prisoners confesses, then he walks free, and the other person gets ten years instead of five. Meaning both have to predict how likely is it for the other prisoner to cooperate with the police.
That's when Nash Equilibrium the Nobel prize-winning theory becomes relevant idea in game theory. It describes any situation where all of the players in a game are trying to secure the best possible strategy given the strategies of all of the other participants. Based on Nash equilibrium the dominant tactics for each prisoner is to confess, because doing so will minimize the average number of years they might expect to stay in prison.
But if both prisoners confess, they're sentenced to 5 years each in prison which is higher than if both prisoners choose to deny their involvement in the crime. Meaning by considering self-interest alone, both prisoners make themselves worse off.
However, even if both prisoners deny the crime, they still have an incentive to cheat one another and confess, thereby reducing their time in prison.
The equilibrium in the Prisoner’s Dilemma takes place when individual players take the best possible action for himself given the action of the other player. After considering every single possible scenario, it's clear to say that the two criminals could do better by both denying – except once the plot sets in, each has an incentive to cheat.
Briefly, in the prisoner's dilemma, the reward for defecting is greater than to cooperate which itself brings a better outcome than mutual defection which itself is better than the becoming a victim of someone else's disloyalty.
With this in mind, how can an individual use such theory in real life?
Naturally, Game theory revolves around the possible outcome that comes from making different decisions. For instance, if an owner of a beautiful piece of art is looking to sell and his motive is to get the most money possible, then the first thing to consider is to decide on what type of auction he needs to use. According to William Spencer Vickrey, a professor of economics and Nobel Laureate from Columbia University, recommends using a second-prize auction to collect the highest possible amount of money. The reason behind this choice rests in the way this sort of auctions uses the final price for the bid. So participants bid and the highest bid wins, one may argue that's exactly how any auction works. However, the highest bid does not pay the highest bid price; rather the bidder pays the second highest bid price. Thus an incentive is created for participants to keep bidding higher and in return, they tend to spend a lot more than they would in a regular auction. By creating this incentive, participants often pay more than they initially anticipated. By following this strategy, the seller is forcing the bidder to spend more and in return earn more for this piece of art.
The use of game theory is almost present in every situation, business or personal. What makes a difference is to have a sense of awareness about the choices because if one individual is not aware, then the opponent has an easy win to secure if the timing is chosen carefully.

Another strategy which is reasonably present in all computer applications: "the Optimal stopping theory". What makes it valuable is that it deals with real situation every time simply because "Time" or "Timing" is the main factor for this theory. It is concerned with the problem of choosing the best possible time as Abraham Wald a Hungarian mathematician states. " Take a given action based on sequentially observed random variables to maximize an expected payoff or to minimize an expected cost. Problems of this type are found in the area of statistics, where the action was taken may be to test a hypothesis or to estimate a parameter, and in the area of operations research, where the action may be to replace a machine, hire a secretary, or reorder stock,". Sequential Analysis (1947).
What Abraham Ward is explaining here that Optimal stopping problems are concerned with the control of random sequences in a statistical decision. The reason for this concern is that people desire to know the optimal instant break off playing a game like a poker player in a casino, or in other examples such as: should a seller of a house accepts an offer from a buyer or wait for a better offer.
Problems of this nature are quite exciting and attractive. This formula can be used to find love as well. For simplicity sake, consider a man looking to get married, there are potential 25 women to meet. Mathematically speaking he needs to reject the = 5 so reject every single one of the first five ladies no matter how good they are and rank them based on their qualities. Select the best out the five ladies and use her as your benchmark, from there every time you meet a new girl compare her to the best choice of the first five. Once a date after the first five woman is better than the standard point of reference, then it is time to settle down.
= n * 37% meaning with this approach this man improved his chances of finding the perfect woman from 4% to 37%. Even though it may sound far from perfect but 33% chance higher than if a person does it in any other way is unquestionably a huge improvement. Improving our lives and coming up with faster and better ways to solve problems is the main reason we invented computers. It is true that some of the algorithms that computers use may sound or at least appears to be far-fetched from real situation. From the surface, one may argue that such computation has nothing to do with people and their daily routines. One also can debate that computers and their overly complicated numerical equations are only used to measure abstract equations or intensive tasks in various fields, including quantum mechanics, climate research, oil and gas exploration, molecular modelling and such. It is accurate to think that computers are used to simplify our work in a complex environment, but they are not limited to just that. What is also very true is that we don't have to use computers directly to benefit from them.

From the previous examples in this paper, individuals can apply algorithms used initially by computers to simplify or solve a problem in a primary, daily situation, all what is needed; a clear understanding of when how and where people can use an algorithm, what to expect, and most importantly the limitation of their values or solutions. As an example of restriction in the case of finding a girlfriend would be 37%, so the chances of having the best woman possible are around that number. It is not 100 percent, but it is also better than the first tiny fraction of 4% chance of finding the perfect woman. Rules, formulas and procedures are written in code give computers the power to make the best possible decisions quickly and efficiently. Therefore, humans should adopt some of those strategies for the betterment of their own lives. One of the main benefits is that applying some of the strategies used by computers can also help save people from particular kind of emotional cost. By knowing that a person is following a suitable algorithm, it could assist in reducing the anxiety level, especially when not sure or clear what is the right thing to do. But knowing that by following a process then some guarantees are given, consequently minimize the level of regret or anxiety. When faced with a hard problem computers break them into smaller pieces by allowing approximations, or a chance and by doing so they produce a decent solution. Computers fail all the time, what is essential is to understand and be willing to settle for a "good" solution. Nothing guarantees the perfect outcome so if human follow those rules we still fail most of the time, but that's all a person can do. We can't control every outcome but only processes and the willingness to choose the best possible choice available at that time. That is precisely what being rational means.
Whether the United Kingdoms is using game theory right at this moment while trying to negotiate the best possible terms once it leaves the European Union, or a college student is trying to find a parking without having to pay for the monthly space by using the optimal stopping approach. All are using some sort of algorithm to help us achieve more, be efficient and reduce wasted time. The real question is what algorithm are you willing to learn and implement today?

By F Gritli