Game Theory, Gamble and Darwin. Ali Hussein, Logan Libonatti, Salvatore Mitrano. Nova Southeastern University

Transcription

1 Running head: GAME THEORY, GAMBLE AND DARWIN 1 Game Theory, Gamble and Darwin Ali Hussein, Logan Libonatti, Salvatore Mitrano Nova Southeastern University Probability theory is basically common sense reduced to calculation; it allows to measure exactly what the enlightened minds feels as a sort of gut instinct without realizing it... It 's remarkable how this science, which began with studies of gambling, it is elevated to the most important objects of human knowledge. -Blaise Pascal

2 GAME THEORY, GAMBLE AND DARWIN 2 Abstract Gambling is an activity that people have participated in for thousands of years. For many, it s a fun thing to do when you have the opportunity, and for others it s an addiction to the thrill of games like poker or just a regular slot machine. In this review we take a look at the reasons people delve into gambling and how it relates to our evolutionary development. Because of man s inability to process risk when they see large odds, they often believe that it can happen to them, which affects some individuals more than others.

3 GAME THEORY, GAMBLE AND DARWIN 3 Table of Contents I. Game Theory, Gamble and Darwin 1. Title Page Abstract Table of contents Literature Review... 4 a. Introduction... 4 b. What is it a game?... 4 c. Gambling... 5 d. The probability and the risk of a gamble... 7 e. Darwin... 8 f. Conclusions Probability Game: FindStars a. Introduction b. Rules & Specifications c. Basic Probabilities d. Advanced Probabilities References... 20

4 GAME THEORY, GAMBLE AND DARWIN 4 Game Theory, Gamble and Darwin Before we begin, let us say something about Game Theory and the misunderstanding that often this discipline brings with it, given the name it bears. Despite numerous applications since the remote past, Game Theory is a branch of mathematics that is very "young". Only in recent decades it has become a relatively stable configuration, even though it still continues to develop and change. Its real date of birth can be traced back only to the beginning of the 20 th century, when the Hungarian mathematician Von Neumann published an research article titled the theory of board games. For the first time, he mathematically represented the rules of table games, formally defined the concept of strategy, and demonstrated that there is a solution for all antagonistic games, which are all the games with opposing players and which the of the sum of winnings is constant. Therefore, Game Theory is so called because it was formalized for the first time just by describing the table games. However, its main focus is on what the table games represent in every day life: strategies and interactions between individuals. This is why, after the publication of this paper, Game Theory has found numerous applications in the social sciences, in evolutionary biology, as well as political analysis and economics. What is a game? To explain how game theory is related to gambling, we must start from the beginning and understand what we habitually refer to as a "game. In the course of human thought, from Aristotle to Plato, up to mathematicians and statisticians, we may encounter many definitions of "game". A particularly interesting is that given by Huizinga, Rector of the University of Leiden in 1933, when he wrote the paper "The limits of the game in cultures". Its intention was to give

5 GAME THEORY, GAMBLE AND DARWIN 5 both a comprehensive and exhaustive definition of "game", and demonstrate the importance of its role in the development of civilization. Going even further back in time, it is significant to see how the birth of more civilized cultures, such as Greeks and Romans, lead to two main types of games. Those games that derives for the word Agon, such as games with precise rules and competition, and those related to the Latin world Alea, which means dice. These last types of games are ruled by Fortune, an impersonal entity that does need neither moral nor skills whatsoever. In games governed by the Alea player relies totally on luck. The challenge is to win, not so much against an opponent, but against the fortune itself. Thus, these games have linked, as degenerative form, superstition, which is the idea that the player could affect chance in some way or another. Hence, the use of talismans, prayers, invocations and divination, lead to multiple chance games related to cards and dices. Gamblers have been the inspired, over the centuries, of notable writers, one of them Dostoevsky. Dostoevsky started writing, ironically, due the economic necessity caused by gambling debts. One of his most famous novels is "The Gambler". In this novel, the writer analyzes the gambling in all its forms and with different types of players, from the rich European aristocracy, the poor souls who play all their belongings, allowing us to understand that the problems linked to the degenerative aspects of the game have not improved over the centuries. Gambling The Alea games, as we have said before, can be seen as a way to take revenge on other aspects of our lives, but at the same time, for those who manage casinos or for the government, these games are simply a source of income. However, from a regulatory point of view it is

6 GAME THEORY, GAMBLE AND DARWIN 6 important to identify which games are made of chance and which are not, in order to classify what can be legally played everywhere and which can be played only in special places, such as in casinos. To create a clear classification of which games are considered chance games or not, we will analyze a game where it is widely believed that the end result depends exclusively on the ability of the two players: chess. What are the characteristics of this game? First of all, for its entire duration, the players are perfectly aware of the actual situation on both sides, in the sense that there is not something known only to one of the players and not the other. A second important aspect is that, in the game of chess, the elements that depend on the case turn out to be negligible or null, in the sense that in a game between friends the assignment of who starts the game is random and in official tournaments each of the two players play the first move at least once. The only thing that is left to the chance is the unknown strategy of each player, including the pitfalls, the diversions and the sacrifice of a few pieces, which cannot be predicted ahead. Therefore, precisely from these observations, a group of researchers of game theory proposed a mathematical definition of the concept of skill players (Dreef, Borm and Genugten). It is defined by considering two parameters: the degree to which the outcome of the game is influenced by the players (learning) and the degree to which the outcome depends on the aspects due to chance (randomness). Learning is defined as the increase of the winnings that an experienced player is able to attract. For example if the skills improve the performance of a player who is able to develop strategies, even sophisticated ones, which allow him to win more than a beginner. Instead, randomness is defined as the increase in payouts for a player who not only know the rules of the game, but who also knows what may depend on the case.

7 GAME THEORY, GAMBLE AND DARWIN 7 The probability and the risk of a gamble Thus, the games whose outcome is strongly, if not exclusively, determined by chance are called gambling. This category includes all games found in casinos, all public bets (Lottery, Scratch and Win, and Sweepstakes), various games with dice, as well as "Christmas Games" (Bingo, merchant at the fair, Blackjack, and Plate). Even some TV Shows are based on a game of chance. However, not all games involving random elements are considered gambling. Bridge or even Poker leave ample room for the skill of the player to try to balance the randomness of the distribution of the cards. In gambling games it is expected the presence of one or more players and a dealer. The dealer proposes to players to gamble on an "event" that is well defined expected to happen, by paying money to play the game. If the player guesses, which occurred the event on which he bet, the dealer pays the player the previously agreed amount, otherwise appropriates the amount of the bet. In games of chance against the dealer, then the player has the sole discretion to determine the amount of bet and the type of bet, as well as, of course, whether to play or not. The player cannot do anything else. Inside of gambling game the player can also do further classification on the basis of a parameter called the hazard coefficient. The hazard coefficient is calculated as the probability that the player has to win, multiplied by how much the player could win, less the chance the player has of losing multiplied by what the player bet. An hazard coefficient greater than zero means that the bet is convenient (obviously does not mean that you will always win. It means that, by repeating the bet several times, on the "long" run you will earn money), whereas if the coefficient is negative, if the bet is repeated in the long run, we will surely lose money (as before does not mean that you will always lose. it means that, by repeating several times the same bet in the long run the player will definitely lose money).

8 GAME THEORY, GAMBLE AND DARWIN 8 The issue is quite clear: people cannot "manage" games of chance; they fascinate people and have a hold on them, because they are not used to reason with large numbers and probabilities. And this brings people to often make wrong choices. This happens because we are what we are, because of evolution. Darwin The reason why people continue to play in spite of these insignificant probabilities of victory is that our brains are built to deal with events on a time scales radically different from those that characterize our evolutionary change. We are equipped to evaluate processes that require completing in seconds, minutes, or years, at most, decades. Therefore our system of judgment is based on skepticism and on the theory of subjective probability and is exposed to very large margins of error, being tuned to a curious irony by evolution itself, to work within a lifespan of a few decades. Probably, the events that we commonly call miracles, both religious and those related to gambling are part of a range of natural events more or less improbable. The only miraculous thing would be the coincidence between what a person asked (to be struck by lightning) and being really struck by lightning. But lets be careful: coincidence means nothing but multiplied improbability. The probability that a person is struck by lightning within one minute of their life span is perhaps from 1 million to 10 million, but also the probability that a person ask to get a lightning strike at a particular time is very low. To calculate the joint probability of this coincidence, we must multiply the two separated probabilities. Roughly, we can say that the probability of this happening is one in 250 billion. Thus, even though the probability is very low, we were able to calculate it. But this dose not means that it is zero.

9 GAME THEORY, GAMBLE AND DARWIN 9 Thus, this probability can happen, if we consider the 7 billion people living on this planet. Therefore, even something very unlikely to happen like a lightning strike could happen in the context of the 7 billion of people. As a matter of fact, considering the many billions of people that such a thing could happen, although unlikely, the coincidence ends up not being really as great as it might seem at first glance. In other words, the global village in which we live, allows us to observe things highly unlikely with a frequency apparently much greater than the real one, and that's because we evolved in the context of a few tens or at most hundreds of people, therefore our brain is not able to "understand" and "imagine" a number of people so high. Everything seems like it's happened to our neighbor and, for this, it seems much more likely than it actually is. It is likely that our ancestors had no need to deal with dimensions and times outside the narrow range of practical needs of daily life, so our brain never developed the ability to imagine things so large or so small. Conclusion Evolution then has equipped our brains with a subjective consciousness of risk and improbability suitable for creatures that have a life span of less than a century. When you have to deal with situations in which our perception is overwhelmed by sizes that we are not used, gambling is one of these situations, it is more rational to realize that the sudden change of life, linked to miraculous gambling winnings, are practically null. Paradoxically, these activities are likely to turn their stay in this world, and not only ours but also of the people who share their lives with us, in a living hell. Where we see a society addicted to gambling or miraculous prediction that are very unlikely to happen.

10 GAME THEORY, GAMBLE AND DARWIN 10 Probability Game: Find Stars In Find Stars, the goal of the game is simply to find the two stars hidden on the field and match them together. It s a memory game, but the outcome is mostly determined by finding the stars the soonest, which is mostly through chance. Once you find the two stars and pair them together, you get a score depending on the number of rounds it took you to do so, and the number of hearts that you have. You start with 30 hearts, so you have a minimum of 30 tries to match the stars together. Hidden throughout the field are other cards that either lower the round you are on, increasing your score, or increase the number of hears you have, which increase the amount of attempts you get, as well as increasing your score. Figure 1

11 GAME THEORY, GAMBLE AND DARWIN 11 Rules & Specifications The Cards: TRY AGAIN CARD: these are the most numerous cards of the game. They have no function but to make the player lose a life and make the player s round increase. + LIVES CARD: Finding these pairs of cards will allow the player to recover some lives lost during the game. There are five different types of pairs, each one allow adding a different amount of lives. The minimum amount of lives the player can add is 1 and the maximum is 5. The player round will increase by 1 when he/she combines these cards, but the player will not loose any lives for that round. - ROUND CARD: This type of card allows you to decrease the number of rounds, thus giving you the opportunity to improve your position in the ranking. The minimum amount of round you can subtract is 1 and the maximum is 3. The player will loose a life when he combines these cards, but he/she round s will not decrease. STAR CARD: These cards are the most important cards of the game. Finding both of them will stop the game and the player will be place in the ranking board. The lower the number of rounds and the higher the amount of lives, higher is the player ranking score. The aim of the game is to be able to find both cards STAR in less possible rounds with the most amount of lives. The lower the number of rounds, the higher will be the player s chances to rank on the top players.

12 GAME THEORY, GAMBLE AND DARWIN 12 Basic Probabilities Let 𝑃 𝑠𝑡𝑎𝑟𝑠 = 𝑃(𝐴) 𝑃 𝑇𝑟𝑦 𝐴𝑔𝑎𝑖𝑛 = 𝑃(𝐵) 𝑁 = 60 Number of stars=2 Number of Try Again = The probability that I will win from the first move. 2 1 𝑃 𝐴𝐴 = = The probability that I will win on the second move. 𝑃 𝐴𝐵 𝐴 + 𝑜𝑟 𝐵𝐴 𝐴 + 𝑜𝑟 𝐵𝐵 𝐴𝐴 = ! = #$ The probability that I will win on the third move. 𝑃 𝐴𝐵 𝐵𝐴 + 𝑜𝑟 𝐵𝐴 𝐵𝐴 + 𝑜𝑟 𝐴𝐵 𝐵𝐵 𝐴 + 𝑜𝑟 𝐵𝐴 𝐵𝐵 𝐴 + 𝑜𝑟 𝐵𝐵 𝐴𝐵 𝐴 + 𝑜𝑟 𝐵𝐵 𝐵𝐴 𝐴 + 𝑜𝑟 𝐵𝐵 𝐵𝐵 𝐴𝐴 = 𝑃 2 𝐴𝐵 𝐵𝐴 + 4 𝐴𝐵 𝐵𝐵 𝐴 + 𝐵𝐵 𝐵𝐵 𝐴𝐴 = = = The probability that I will win on the fourth turn. 2 𝑃= = The probability that I will win on the fifth turn. 𝑃= =

13 GAME THEORY, GAMBLE AND DARWIN 13 You can see the pattern. The first and second terms increased by 2. Then last probability will look. 29. P =! #$ 30. P =! #$ = = We set up a table. Number of the last turn. Probability

14 GAME THEORY, GAMBLE AND DARWIN Σ= If the 30-th move will be as follows, it means that I lost. Because the 31st move is not available to connect the results of the star. It will look as follows: P(AB*BB* *BA+(or)BA*BB* *BA+(or)BB*AB* *BA+..+ BB*BB*.*BA*BA)=1-Σ=1-0, =0, I think that the probability of +3, +5 influence will be weak. Accidentally rarely possible to combine them, and especially combined - must spend 1 turn. -2, -3, Very rare. I can calculate the probability, but, basically, it is possible to navigate in the table.

15 GAME THEORY, GAMBLE AND DARWIN 15 Complex Probabilities Assume that a player adheres to the following strategy: on each step he chooses different pairs of cards and remembers it. If he find for example two cards -1 Round earlier than two cards with stars on different steps he ll open them together on the next step to decrease the number of Round. So this type of cards helps a player to increase the probability of winning the game. First, I introduce the notation of number of ways to divide n elements into pairs. So we have ways to choose the first pair and that n 2 elements left. Next, we choose a pair from n 2 elements. We have C ways to do this. Then n 4 elements left. And so on while we have not less than 4 elements. Then we choose 2 elements from 4 and 2 elements left so it ll be the last pair. Pair n = C!! C!!!! C!! = n! (2!)!! There C!! =!! is the number of combinations of k elements from n elements.!!!!!! Next, consider different situations that can occur throughout the game and introduce new notations for them. 2 n 2 So, let s start with the probability of finding two stars on ( k ) + 1' thstep. Divide this probability on two parts, because a player can find two stars separately on first k steps k 1' th k + 1' th and than just open them together on ( + ) step or he can find them together on ( ) step. The number of all possible variants of getting k + 1 pairs of cards from 60 is C!!!! Pair(2 k + 1 ) Consider the first part of this probability. On first k steps we must choose two stars but not get two cards that decrease number of rounds (this cause ll be described further). And on ( k ) + 1' thstep we ll just open together two cards with stars. So we must choose other 2k 2cards for first k pairs from =55 cards (we must choose two stars and less then two cards decreasing number of rounds) 2 C n

16 GAME THEORY, GAMBLE AND DARWIN 16 and divide them into pairs in such way that we haven t two stars together on some step (we have kpair ( 2 ( k 1)) ways to do that because we can choose the place for this pair in k ways and divide other cards on k 1 pairs). So you can see the final form of this part of probability in first term of the following formula. Now consider the second part of the probability when a player finds two stars together on ( k + 1) ' th step. So he must choose 2k cards for first k pairs and divide them into pairs. Summing these two cases we have the probability.! 𝑃!! 𝑘+1 =!! 𝐶!!!! ((𝑃𝑎𝑖𝑟 2𝑘 𝑘𝑃𝑎𝑖𝑟 2 𝑘 1 ) 𝐶!!!! 𝑃𝑎𝑖𝑟(2 𝑘 + 1 ) +!! 𝐶!! 𝑃𝑎𝑖𝑟(2𝑘 2) 𝐶!!!! 𝑃𝑎𝑖𝑟(2 𝑘 + 1 ) Next, consider the probability of getting separately two cards decreasing the number of rounds earlier than two stars. Arguing similarly we get the following formula.! 𝑃!! 𝑘+1 = 𝐶!!!! ((𝑃𝑎𝑖𝑟 2𝑘 𝑘 𝑘 1 𝑃𝑎𝑖𝑟 2 𝑘 2 ) 𝐶!!!! 𝑃𝑎𝑖𝑟(2 𝑘 + 1 ) + 𝐶!!!! (𝑃𝑎𝑖𝑟 2𝑘 𝑘𝑃𝑎𝑖𝑟 2 𝑘 1 ) 𝐶!!!! 𝑃𝑎𝑖𝑟(2 𝑘 + 1 ) The probability of getting together two cards decreasing the number of rounds earlier than two stars. Arguing similarly we get the following formula.! 𝑃!! 𝑘+1 = 𝐶!!!! 𝑘 𝑘 1 𝑃𝑎𝑖𝑟 2 𝑘 2 + 𝐶!!!! 𝑘𝑃𝑎𝑖𝑟(2(𝑘 1) 𝐶!!!! 𝑃𝑎𝑖𝑟(2 𝑘 + 1 ) Next, consider the probabilities of getting separately two and three pairs of cards decreasing the number of rounds earlier than two stars.

17 GAME THEORY, GAMBLE AND DARWIN 17! 𝑃!!!! 𝑘+1 = 𝐶!!!! 𝑃𝑎𝑖𝑟 2𝑘 𝑘 𝑘 1 𝑃𝑎𝑖𝑟 2 𝑘 2 + 𝐶!!!! 𝑃𝑎𝑖𝑟 2𝑘 𝑘𝑃𝑎𝑖𝑟 2 𝑘 1 𝐶!!!! 𝑃𝑎𝑖𝑟 2 𝑘 + 1, 𝑖, 𝑗 = 1,3, 𝑖 𝑗! 𝑃!!!!!! 𝑘+1 =!!!!! #$!!!!!!! #$!!!!!!!!! #$(!!!! ) +!!!!! #$!!!#$%!!!!!!!!! #$!!!! The probability of getting together two or three pairs of cards decreasing the number of rounds earlier than two stars. The second index the top means number of pairs getting together. 𝑃!!!! 𝑘+1 = 𝐶!!!! 2𝑘𝑃𝑎𝑖𝑟 2 𝑘 1 + 𝐶!!!! 2𝑘 𝑘 1 𝑃𝑎𝑖𝑟(2 𝑘 2 ) 𝐶!!!! 𝑃𝑎𝑖𝑟 2 𝑘 + 1!! 𝑃!!!! 𝑘+1 =!! 𝐶!!!! 𝑘(𝑘 1)𝑃𝑎𝑖𝑟 2 𝑘 2 + 𝐶!!!! 𝑘 𝑘 1 (𝑘 2)𝑃𝑎𝑖𝑟(2 𝑘 3 ) 𝐶!!!! 𝑃𝑎𝑖𝑟 2 𝑘 + 1 𝑃!!!!!! 𝑘+1 = 𝐶!!!! 3𝑘𝑃𝑎𝑖𝑟 2 𝑘 1 𝐶!!!! + 3𝑘(𝑘 1)𝑃𝑎𝑖𝑟(2 𝑘 2 ) 𝑃𝑎𝑖𝑟 2 𝑘 + 1!! 𝑃!!!!!! 𝑘+1 = 𝐶!!!! 3𝑘(𝑘 1)𝑃𝑎𝑖𝑟 2 𝑘 2 + 𝐶!!!! 3 𝑘 1 𝑘 𝑘 2 𝑃𝑎𝑖𝑟(2 𝑘 3 ) 𝐶!!!! 𝑃𝑎𝑖𝑟 2 𝑘 + 1 𝑃!!!!!! 𝑘+1 𝐶!!!! 𝑘(𝑘 1)(𝑘 2)𝑃𝑎𝑖𝑟 2 𝑘 3 + 𝐶!!!! 𝑘 𝑘 1 𝑘 2 𝑘 3 𝑃𝑎𝑖𝑟(2 𝑘 4 ) = 𝐶!!!! 𝑃𝑎𝑖𝑟 2 𝑘 + 1

18 GAME THEORY, GAMBLE AND DARWIN 18 There i, j = 1,3, i j. Now we can go to the calculation of the probabilities of ending the game in k rounds. First, introduce some notations. The index in P j ( k) means the difference between number of steps the player has done and number of rounds he has after that. P k P k ( ) = 3 ( ) 0 P ( k) = P ( k + 1) P ( k) = P ( k + 2) + P ( k + 2) P ( k) = P ( k + 3) + P ( k + 3) + P ( k + 3) P ( k) = P ( k + 4) + P ( k + 4) P ( k) = P ( k + 5) + P ( k + 5) + P ( k + 5) P ( k) = P ( k + 6) + P ( k + 6) + P ( k + 6) P ( k) = P ( k + 7) P ( k) = P ( k + 8) P ( k) = P ( k + 9) If P ( k) j doesn t exist ( k j 30) P j ( k) = 0. + > assume So, the probability of ending the game in k rounds. ( ) P k 9 = Pi ( k) i= 0 And the probability mass function ( ) F k k = P( k) i= 1

19 GAME THEORY, GAMBLE AND DARWIN 19 Next, consider the basic probabilities of finishing the game on ( k ) + 1' thstep. We also divide it into two cases: when a player finds two stars on first k steps separately and when he finds them together on k + 1' th step. ( ) In first case he must choose also another 2( k 1) cards from 60-2=58 cards (without two stars) to make first k pairs of cards but without the case when two stars are opened together (he opens them on k + 1' th step). ( ) In second case first k pairs are made without two stars (from other 2k cards). Summing these probabilities we have P k + 1 = C!!!! (Pair 2k kpair 2 k 1 + C!! Pair(2k) C!!!! Pair 2 k + 1 And the probability mass function by definition ( ) F k k = P( k) i= 1

20 GAME THEORY, GAMBLE AND DARWIN 20 References Neumann, J. Von. (1928). Zur Theorie der Gesellschaftsspiele. Mathematische Annalen 100 (1928): Retrieved from M. Dreef, P. Borm and B. van der Genugten. (2001, December 20). A New Relative Skill Measure for Games with Chance Elements. Retrieved from Alfthan, E. (2007, May 21). Optimal strategy in the childrens game Memory. Retrieved from: Mazalov, Vladimir. (2014). Mathematical Game Theory and Applications. Wiley. Retrieved 2 December 2015, <