Mathematics that leads us to become better individuals, society and world

We learned about mathematics ever since we started going to school. We studied it every year as a subject, as saw it as it was – a subject taught in school. For us, mathematics was basically a scientific discipline that existed only in the realms of the four walled classrooms. We never really saw mathematics as a discipline that existed outside the classroom. Sure numbers and arithmetic helps us from time to time, but that’s it. The rest of mathematics has been locked up as we left it in the classrooms.  I mean, why should we bother with mathematics? We live in a very competitive world that teaches us that if you can’t keep swimming, sooner or later you would have to sink in the deep abyss of the ocean. We disregard mathematics because to us, it has no aid in surviving this competitive world. To us, it is a mere subject that deals with numbers and thus can never aid us in deciding which path to take in order to survive.

But the book “Rock, Paper, Scissors” tells us the contrary. It tells us that mathematics do exists outside the classrooms. It tells us that mathematics, especially game theory, can help us in the different situations of our lives, especially in decision making.

Rock Paper Scissors: Game Theory in Everyday Life is a mathematical book written by the physicist and Nobel Prize winner, Len Fisher.  This book is an introductory course about what game theory is and how it’s various situations can be applied to problems that we deal in the real word.  The topic on game theory varies widely and can be applied to everyday situations. Fisher also cites actual incidents where these topics can be applied. The book also works through various solutions to many of the common dilemmas that people encounter when dealing with other people. And these solutions aren’t just mere opinions of how these problems can be solved; rather they are real solutions with mathematical proofs.  (Gotch, N., 2010)

Basically, this book explains to us what game theory is and how this game theory can be utilized to promote cooperation among individuals, communities and even nations.  (Kaara, 2009) Right from the beginning, Fisher says that this book is concerned with the part of game theory that “concerns cooperation rather than confrontation, collaboration rather than competition.” According to one of the reviews: “Accordingly his conclusion, that “individuals can make a difference,” applies mostly to daily collective active problems that annoy individuals, rather than the longer-scale ones that drastically compound human suffering over the course of decades.” (socialventures.org, n.d.) Thus he relates the concept of game theory to be utilized in a way that can promote an environment that favors cooperation and collaboration. He stresses in the book that if we could find ways to trust each other, we could find win-win solutions to many the serious problems that we all face.

First, the book tells us what game theory is concerned about. Game theory, according to this book is the study of strategic decisions in situations where the results of strategies depend on other participants’ strategies. (em-and-emm.blogspot.com, 2011) It is concerned basically with decision making when the decision making individual/s are confronted with competitive situations, especially when they have limited information about the other party’s choices. It is also said that for every competitive situation, there is a point called a Nash Equilibrium, in which parties cannot change their course of action without sabotaging themselves.  And according to Fisher, situation can actually be arranged in such a way that the Nash Equilibrium provides the best possible outcome for everyone. And to relate it to cooperation, fisher examines how social norms and people’s view of fair play can effect into cooperative solutions rather than competitive ones. (publishersweekly.com, 2008) This means that game theory looks at how we maximize our gain in competitive situations because game theory assumes that it is self-interest that controls how people decide. (Dixon A., 2008)

The book also demonstrates the limits of game theory, saying that “what is best for you isn’t always best for everyone else” and this discrepancy can undermine your own self-interest. (Dixon A., 2008) Aside from this, the book also looks at game theory as a concept that investigates the motives and dilemmas of social interactions relative to selfishness and cooperation. Thus, we know game theory we can then increase our chances of finding and satisfying resolutions by adopting new strategies.  (Benjamin, 2010)

In the book, Fisher also gives tips that can be applied to some situations that we might encounter. These are (these are list in a form of verbatim from a review by Benjamin on October 22, 2010 in goodreads.com)

1. Keep the same strategy if you’re winning, shift strategies if you lose.
2. Bring a third player in. They can be a known negotiator or a known cheater – either way it helps.
3. Set up reciprocity. Knowing that you’ll deal with people after a conflict can increase the incentive to cooperate.
4. Limit future options or provide incentives. This shows that you are committed to the best possible outcomes.
5. Offer trust. It’s simple, but it can be effective.
6. Create a situation from which neither party can escape from without loss.
7. Use side-payments to maintain cooperation.
8. Know the seven deadly dilemmas and avoid the worst outcomes:
a. The Prisoner’s Dilemma – all must cooperate or all fail.
b. The Tragedy of the Commons (a series of Prisoner's Dilemmas) –self-interest prevents cooperation despite impending long-term failure.
c. The Free Rider problem - people taking advantage of a community resource without contributing to it.
d. Chicken/Brinkmanship - each side tries to push the other as close to the edge as they can, with each hoping that the other will back down first.
e. The Volunteer’s Dilemma - someone must make a sacrifice on behalf of the group, but if no one does, then everyone loses out.
f. The Battle of the Sexes - two people have different preferences, but each would rather share the other's company than pursue their own preference alone.
g. Stag Hunt - cooperation between members of a group gives them a good chance of success in a risky, high-return venture, but an individual can win a guaranteed but lower reward by breaking the cooperation and going it alone.
9. Work to create transparent processes that are inherently fair.
10. Favor smaller groups, it’s easier to foster trust and cooperation
(Benjamin, 2010)

And lastly, the book is not only rich in content but in examples as well, as Fisher uses these examples to demonstrate the points stated. We see how Fisher uses simple anecdotes, drawing experiences from his personal life and others’ to illustrate how game theory works. These experiences ranges from the experiences he has with drivers, women he meets while in the bar, his family and relatives. (socialventures.org, n.d.)

This book is an epitome of an introductory guide to decision making. Though it deals with mathematics, yet it connects this mathematics to real situations that exist outside its realm. It connects mathematics, through game theory, to aiding us in making better decisions for ourselves, for the society and for all mankind.

Individual. (Decision) Mathematics, specifically game theory, aids in individual decision making. The key word here is decision. Through mathematics, we become better decision makers. We may have lots of friends that aid us in decision making, but we have to admit, by the end of the day it will just be us, our selves that we can count on. Especially in the era of competitiveness where everyone wants to be somebody and to step up in the social ladder, we realize that our only ally is our selves. And this book had shown us that mathematics can help us see things clearer. It helps us realize that mathematics is a tool which we can use to view all the possible solutions and the best solution to choose that will not compromise our own position. We realize that mathematics actually helps us survive by becoming individuals that makes decision better.

Society. (Cooperation) Through this book, we are able to see that mathematics can actually unite a society together. We see that through mathematics, there can exist cooperation because in certain situation, it is best that we seek interest that will benefit all the involved parties. If we become better decision makers, we can choose not only to uplift our individuality but also of our society. Let’s admit that we view our society as a group in which we belong, and as part of belonging, we will always choose what is best and what is best for our group. And through this book, we see that mathematics is not only a subject, nor is it only for ourselves but we see mathematics as much bigger, something that can uplift our society. We see it as something that can help us improve our society through the cooperation that is reflected in our decision making. The key here is cooperation. Because we are better decision makers, we will always pursue cooperation. That way not only will the self-interest be met, but of the group in which we belong with as well.

All mankind. (Unselfishness) This book aims for collaboration and cooperation instead of competitiveness. And though we live in a very competitive world, none of us actually wants it. We were just born in a society that views competitiveness as a drive to survive. But do we really want it? The answer is no. We would all want something less extreme, something that doesn’t make us go crazy and something that will allow us to become less selfish. Let’s face it; living in a competitive world opens the door to selfishness. We all want to win, nobody wants to lose. And sometimes we even forget our humanity for the sake of idea of winning and being on top. But through this book we can see that it doesn’t have to be that way. Mathematics is a tool to aid us not to become selfish but to be better. That if we become better decision makers, we would choose to cooperate and if we choose to cooperate we become unselfish for we see that by being unselfish we help ease the pressure and the tension that builds up because of the concept of competition.

“Swim or sink.” That’s what we always here. We have to swim in order to survive this very competitive world. But swimming doesn’t have to lead us in a direction of becoming bad decision makers or unselfish people. Rather if we really want to swim, we will have to take the courage to admit to ourselves that sooner or later we would get tired and would sink eventually. If we really want to swim we have to admit that we can do it better if we choose better and if we cooperate better. This means that we have to look for aid that can help us survive the heavy waves of the ocean. And this aid could be mathematics. But it doesn’t end there for mathematics actually lends itself for us to become better decision makers that will pursue cooperation through being unselfish.

And who knew that the mathematics we have always disregarded as a mere subject in school could teach us not only to become better individuals, society but to become a better world.

Reference
Gotch, N. (2010, April 16). Rock, Paper Scissors: Game Theory in Everyday Life by Len Fisher [Review of the book Rock, Paper Scissors: Game Theory in Everyday Life].  goodreads. Retrieved February 28, 2014 from http://www.goodreads.com/book/show/3106009-rock-paper-scissors

Benjamin. (2010, October 22).Rock, Paper Scissors: Game Theory in Everyday Life by Len Fisher [Review of the book Rock, Paper Scissors: Game Theory in Everyday Life].  goodreads. Retrieved February 28, 2014 from http://www.goodreads.com/book/show/3106009-rock-paper-scissors

Kaara. (2009,July 18). Rock, Paper Scissors: Game Theory in Everyday Life by Len Fisher [Review of the book Rock, Paper Scissors: Game Theory in Everyday Life].  goodreads. Retrieved February 28, 2014 from http://www.goodreads.com/book/show/3106009-rock-paper-scissors

Book Review: ‘Rock Paper Scissors — Game Theory in Everyday Life’. (n.d.). In Social Ventures for Sustainability. Retrieved February 28, 2014 from http://socialventures.org/book-review-rock-paper-scissors-game-theory-in-everyday-life/

Rock, Paper, Scissors: Game Theory in Everyday Life. (2008, September 22). In Publishers Weekly. Retrieved February 28, 2014 from http://www.publishersweekly.com/978-0-465-00938-1

Rock, Paper, Scissors by Len Fisher. (2011, June 13). In Em and Emm Expound on Exposition Emma and Emily review books!. Retrieved February 28, 2014 from http://em-and-emm.blogspot.com/2011/06/rock-paper-scissors-by-len-fisher.html

Dixon A. (2008, December 1).  Book Review: Rock, Paper, Scissors [Review of the book Rock, Paper Scissors: Game Theory in Everyday Life].  Greater Good The Science of a Meaningful Life.  Retrieved February 28, 2014 from http://greatergood.berkeley.edu/article/item/book_review_rock_paper_scissors

Let the Game Begin!

Let the game begin in 10…   9…   8…   7…   6…   5…   4…   3…   2…   1…

RRRrrrriiiinnnngggg!!!!!!!!!!!
           The sound of the alarm clock awakens you. It says 8:00 am. You immediately jump off  the bed 

           and hurriedly scamper to the comfort room. After a 5-minute shower, you put on your garments,

           brush your hair a  little,  run  off  to  the kitchen, grab toasted bread, wear your shoes and race to 

           the  door.  Now,  you’re   ready   to  face  and  compete  with  the  bustling  people  outside  your 

           house and accept the challenges that await your day.

Life is a game. Many people had developed strategies or ways on how to play it. You should make decisions wisely. Speaking of decisions, there is a study of strategic decisions where the results of strategies depend on other participant’s strategies, this is called Game Theory. 

To learn and understand game theory better, let’s take a look at a book entitled “Rock, Paper, Scissors: Game Theory of Everyday Life” by Len Fisher. This book would be a very good book to introduce game theory to beginners, like me, to learn more about game theory. 

The example of game theory used in the book is the famous Prisoner’s Dilemma, a theoretical situation, in which two suspects are arrested and imprisoned for a crime they probably committed. Each of them is questioned separately and offered the same deal. They were asked to confess and if no one does, both of them will be convicted for one year but if both of them confess, they will be again imprisoned, let us say, for at least 3 years. If one of them betrayed the other while the one being betrayed stay silent, the confessor will be freed while the one who remained silent will be put to jail, more than necessary. And then he introduced the Nash Equilibrium made by John Nash. Its idea mainly say that it is a position in which both sided have selected a strategy and neither side can then independently change its strategy without ending up in a less desirable position.

Game theory is everywhere. The author incorporated it with the common events that happen in everyday life. For instance is the wedding. A guy offered two pieces of cake, sliced unequally, to a person. That person chose the smaller one because he/she would feel bad taking the larger slice. The game theories call that sort of overall benefit “utility” in which all the factors had been taken into action. He also introduced the concepts of minimax and fair division. He highlighted it with anecdotes like how he got in trouble while he shooted fireworks when he was still a kid. Minimax means looking at a situation to see how much you might lose and then planning your actions so as to minimize that loss.

The book does not only discuss Prisoner’s Dilemma but as well as Tragedy of the Commons, Free Rider, Chicken, The Volunteer’s Dilemma, Battle of the Sexes, and Stag Hunt. He also associated the famous game played around the world, the rock-paper-scissors game. I was really amazed on how he used it in conflict resolution. I think the reason is that the game has no pure strategy that makes you dominate the others. Thus, situations and games which seem to be at a standstill can be solved by adding strategies and making them to rock-paper-scissors situations.

The rest of the book revolves about cooperation: how we can gain others trust, offer bargain effectively, and then changing the game to avoid being trap of the Prisoner’s dilemma and other unpleasing results.

As what I had said at the earlier part, life is a game. Its either you win or lose and winning depends on how you play it. You should make good strategies and decisions to live smoothly. And in every game that you encounter, you should have a weapon that will help you play it out well. To conclude this, I will leave a famous phrase that I think each one of us is praying for, “May the odds be ever in our favor”.

Major Minus

Life has always been a mystery ride. You never knew where it can lead you or why it takes you there. But behind those rides, life is simply a game you play. A tough game of chances we play for everyday in our lives. An adventure that made us win or lose.

In Rock, Paper, Scissors, physicist Len Fisher demonstrates the game theory of life. Game theory studies how we maximize gain in competitive situations, assuming that self-interest controls why people make decisions (Dixon, 2008). Also, Fisher showed the limits of game theory, such that what’s best for one particular person may not be best for everyone in the society. Take for example, it may be easier for you to cross the road without taking the pedestrian lane, but if everyone followed that, there may be an increase in road accidents and more people would die.

However, on the brighter side, we humans tend to show cooperative behaviors than some of us might assume. Fisher uses game theory to show how cooperative behaviors have evolved to escape the traps created by selfishness, and how cooperation persists in even the most difficult circumstances (Dixon, 2008).

The book consists of various games that serve as models for real-life solutions. For instance, the game “Tit for Tat,” a payback strategy that comes into play when two people or groups are likely to meet repeatedly, showed that people are more likely to cooperate with you in the future if you have cooperated with them in the past: Suppose a person helped a woman carry the boxes to her car. Assuming the two people meet again, the woman will have the incentive to return back the favor by either helping the person out or by treating him to coffee or food. Fisher also concluded that the two main reason people cooperate is by the statement, “do unto others what you would have them do unto you” or by imposing threats to get the work done.

Nevertheless, Fisher’s game theory solutions are all but entertaining and morally educating and idealistic. But what happens if the woman didn't return the favor? Even so, Rock,Paper, Scissors is a wonderfully entertaining and relevant book concerning game theory and the science of cooperation.

I'll Bet Algebra Was Cool

Is algebra important in the real world? How is it necessary? How could algebra be used to buy goods in a store? Duh!  

(Those are just statements of people from all walks of life that when asked how algebra can be essential to our daily life adventure)

     Honestly, algebra has always been a huge part of our lives. From paying our fares when riding a jeepney to calculating the distance traveled. However, most people do not realize that algebra is used almost every day in their lives. Why is that anyway?

    To begin with, first may I introduce to you our group, Fear Busters! Technically, our group aims in removing all your anxieties and fear of algebra so that you could face algebra problems with confidence and a huge smile on your faces. Sounds cool huh? But anyway, let me tell you how our group do the magic.

    For the first session, we organized a game. Actually the reason why we skipped the lecture part is to impart that algebra is really fun and harmless. Through this, students would experience algebra without thinking all the out-of-this-world equations and solutions; rather they would feel that algebra is just a sort of a simple arithmetic. And later, when we proceed to the lecture part, they would not fear the equations and instead look the problems in a more positive thinking.

     For the first part of the game, we created a quiz bowl participated by groups. The quiz bowl consists of all the topics in algebra, from linear equations to quadratic equations. The game has 3 rounds, the easy, average and hard levels. The points garnered by each group would be ranked according to lowest to highest. Once they were ranked, it would determine their number for the second game.

     The second game involves physical and mental strategies but for the fun part, no calculations needed! The objective is simply to find all the letters in the ALGEBRA word in the shortest time possible. The letters were labelled by numbers 1-3, which determine the rank of each group and also to ensure that the groups collect the letters marked for them. So yeah, the winners would be guaranteed with a sweet price!

   Now, for the second session of our Fear Busters day, we delivered our group’s awesomazing presentation. We first presented our impromptu skit tackling all the topics in algebra. Of course, we won’t tackle it in a boring and ruthless way. Remember that our main goal is to let you enjoy algebra as much as possible? So, we made sure you’re getting much of our dear services.  (HAHAHAHA)

     Once we’ve finished presenting the topics in algebra, we put it into debate. The debate was about the importance of algebra in our daily lives. The group divided into pros and cons. The pros claimed that algebra is useful in our daily lives, provided with examples and convincing reasons. While the cons argued that, “No, it has no importance in our daily lives. Stop relating algebra and live the life without it!!!!” It was a tough comparison of ideas, knowing that both groups have its own point. But in the end of the day, it was you, yourself, who will decide if you needed algebra by your side.

    To put it in general, we just want to imply that algebra is reasonably important no matter what application you would use to put into your life. However, it is your decision to use it or not. But, you must also remember that algebra with its advantage can take you to many places and mold you into someone you wanted in the future. Choose wisely. In the end, it’s your own decision that mattered.

       We are just here if you needed some help!

PS: Thank you for cooperating and also for having fun with us! It really meant a lot to us. Arigatou gozaimasu!

Child's Play

Making decisions for yourself could either be difficult or not depending on the situation you’re in but what about making decisions that doesn’t just involve you but other people. Would you choose the way that would completely benefit you but not the other person or would you choose to give way to that person? This is where Game Theory comes in. Through Game Theory, we are able to know how and why people uses different kinds of strategies when faced with this situation. Game Theory studies how people maximizes gain during competitions and the like. Over the years, game theorists has named the Seven Deadly Dilemmas that people encounter. The Prisoner’s Dilemma, Tragedy of the Commons, Free Rider, Chicken, Volunteer’s Dilemma, Battle of the Sexes and Stag Hunt. All of this are in a book called Rock, Paper, Scissors: Game Theory in Everyday Life. A critically acclaimed book written by Len Fisher and author of Weighing the Soul and How to Dunk a Doughnut, books that were named Best Popular Book of the Year by the American Institute of Physics.

            In our world today, everything is designed to deceive our brain so how do you know who to trust? In a television show I watched called Brain Games there was an experiment wherein 2 strangers compete over a briefcase full of cash called Split or Steal. The rules of the game is that when both of them raised the placard Split, they could split the cash. When they both raise the Steal placard, they can’t get the money. However, when one raises the word Steal and the other Split, the one with the word steal gets all the money. The experiment shows that 3 pairs of total strangers split the money. However, there are participants who decided to steal from the other person. This experiment shows that more people today are trusting because in order to survive in this world we tend to default to cooperation hoping that others would do the same. And this is true because in order for our society to prosper, people must learn how to trust and cooperate with one another. However, for me one should not give their 100 percent trust to the people you are not familiar with because you might get disappointed if they do the opposite of what you expect. This is explained in Stewart’s book by the Tragedy of the Commons wherein cooperation produces the best overall outcome.

            Prisoner’s Dilemma shows the different choices one could make when one is trapped in situation. And a Nash Equilibrium shows a choice in which one could not get away from without ending up in bad position. Stewart also presented the two major challenges to cooperation which are “To find a way to reach coordinated agreements” and “To find a way to make people stick to those agreements”. The Tragedy of the Commons is one of the Deadly Dilemmas. It involves different pairs of people which plays a series of Prisoner’s Dilemma. Next is the Free Rider wherein one takes advantage of the people around him/her and not doing anything. The Chicken or Brinkmanship is another Deadly Dilemma that pushes one another to a verge of falling until this person backs down first. The Volunteer’s Dilemma shows that one sacrifices himself/herself in order for the community to gain. Battle of the Sexes, shows that two people has different preferences and that they prefer to go with the other than separating just to get what they want. Stag Hunt is a Deadly Dilemma that gives a better result when done as a group than going alone and breaking the bond.

            All in all, Stewart was able to explain each of this Deadly Dilemmas in a way that the masses would be able to understand and comprehend. And for me, this book is exciting to read and would help me to be cautious in making decisions in the future.

By: DAISIC DE ASIS BELLO

A Review on Rock, Paper, Scissors ( book)
Len Fisher

Rock, Paper, Scissors is a very informative book presenting truths most of the people are  asleep on. From the moment the morning sun kisses our cheeks to the time it leaves for a rest, we face instances and be inside circumstances wherein decision-making is a necessity and self-interest is the hidden and first consideration. Relating daily events and serious issues like the Kyoto protocol (signed by developed countries telling them to reduce carbon emissions) and the Cuban Missile Crisis (which sparked the bloodless yet scariest cold war between US and USSR) to game theory, the author spoke of the omnipresence of choices and the somehow egocentric nature of humans. The word cooperation was introduced as a form of compromise, where the aim of both parties is partially or fully achieved.

            One of the eureka-mode lines in the book is this : the paradoxes and problems come in when a strategy  of cooperation would lead to the best outcomes for all concerned  but each party is tempted to try for a better outcome for itself, only to become trapped by its own greed in an inferior situation, a.k.a tragedy of commons, the first of the deadly dilemmas. Crabs pull each other down while exerting all their might to get to the top without knowing that they are up to an I-regret-this-situation moment. Second is the Free rider problem   when some politicians put farmer’s and worker’s money on their pockets. So timely. The third is Chicken (I, SERIOUSLY, am getting hungry and I can smell the scent of some meat thing dish up in the kitchen). Anyway, the undying issue on Spratly islands and its rightful owner/s illustrates this. Country A sends troops here, Country B there. One, presenting legal documents defending ownership, the other talking of no fictional tales of the island’s very strong importance and existence on their history. Both countries “push the other to the edge as they can, with each hoping that the other will back down first”.

            “If no one volunteers to solve the word problem, I will give it as a quiz.” This is one of the frightening lines that could ever come out of a teacher’s mouth. Then all of the student’s minds would silently shout in unison somebody solve that already, avoiding their teacher’s face and leaning in different directions. Everybody wants the same thing to happen but nobody has the courage to make the move. As the author pointed out, each person hopes that someone else will make the sacrifice on behalf of the group. And the nerd saved the day.

            A radio jock said in his late night program that most men do things or be in places they don’t like but continue to do so in the name of being a boyfriend to a girlfriend. Going to parks, watching romantic movies, wearing cute bunny headbands and doing selfies are some. Most girls do not find watching basketball and wrestling to be time worthy, but they would rather do so to be able to share with each other’s company. Dilemma number 5 :Battle of the Sexes. Meanwhile, after hearing each other’s business promises and knowing about the members’ credibility and good reputation and the assurance of full cooperation, everyone feels the scent of success.This goes to people who are members of a company, corporation and even classroom groupings who were given specific tasks having that burning desire to triumph.

           

One tip the author suggested is to stay if you win, shift if you loose. I remembered the last episodes of ANTM Cycle 20. There were three remaining models left –one female, two male. On the other hand, the eliminated models chose among themselves who they think would win. Team Marvin, consisting of four people wore blue shirts, team Jourdan with only one, wore orange, while team Cory, having the most number of supporters, wore pink. The resident photographer  told the supporters (models) that they can switch sides and win a thousand dollars if the model they are supporting wins. Mike, from Cory’s team decided to choose Jourdan. His decision was right. This 6’1 blonde bombshell won. For the record, she has won a lot of challenges and she has the most number of best photos. So Mike and Chris went home with the money. Well, there are nine other tips he suggested which I believe, are most helpful.

The author’s points was well explained and concrete examples ranging from the most typical event in life  to social, political and environmental happenings was given. Structurally wise. Plain language was used and dummies will not find it too difficult to comprehend. Reading the first sentence may give a sense of boredom, but there are some force that holds one back , and unknowingly, he or she has read many parts and even finished the book already, digesting sold ideas while realizing that a similar and connected event occurred in his or her life. He was able to create a work patterned on his own unique but not that distinct style. His purposes and that is to let people see math in a different, beautiful light was successful.



Glory to God

           

rock, paper, scissors ?? hulat lang naa pa tay exam

Rock, Paper, Scissors

          

           “Rock, Paper, Scissors is a refreshingly informal as well as insightful account of key ideas in game theory. Len Fisher gives many examples, several from his own life, of games that post harrowing choices for their players. He shows how game theory not only illuminates the consequences of these choices but also may help the players extricate themselves from situations likely to cause anger or grief.” 

                                    – Steven J. Brams, New York University, author of Mathematics and Democracy

Because my mind is so clouded, and I don't know how to start this, I decided to search for blogs about this book, One blogger have said that "Rock, Paper, Scissors is a popular science book that connects  game theory to everyday situations and suggests several strategies for achieving cooperation". Fisher shows how the modern science of game theory has helped biologists to understand the evolution of cooperation in nature, and investigates how we might apply those lessons to our own society.

What's great about this book is that it is not that dull to read, and it is not purely about mathematics. Whenever something is being tackled about, it will be more interesting if it would be connected to real life situations, like what Fisher did.

The first chapter of the book describes the Prisoner’s dilemma and gives the negative connotation that the Nash equilibrium is a logical trap. The matrix graphics are not that illuminating either. The second chapter offers a nice introduction to the concepts of minimax and fair division. Fisher illuminates fair division with anecdotes like how he got in trouble as a kid shooting fireworks, and as a consequence had to yield fireworks with his brother. The answer he intuitively arrived to as a kid was what he know realizes was an application of the minimax principle. Chapter three is about seven of the most interesting game theory problems, which Fisher aptly dubs “the seven deadly dilemmas.” Here Fisher offers a great summary of such problems as the free rider issue and the game of chicken. Chapter four is a humorous one, and is about the game “rock, paper, scissors.” It’s amusing how rock, paper, scissors can be used in conflict resolution. The reason is that the game has no pure strategy that dominates the others. Hence, situations and games which seem to be at a standstill can be solved by adding strategies and converting them to rock-paper-scissors situations. Finally, chapters’ five to eight are all about cooperation: how we can achieve trust, bargain effectively, and change the game to avoid the “trap” of the Prisoner’s dilemma and other undesirable outcomes. I won’t go into detail, as the main fun points are similar in nature to the other chapters: the narratives and interesting examples from science.

High Jinks!

              When we were told to make a review about Ian Stewart’s: Cabinet of Mathematical Curiosities, I was quite hesitant for I believe that this book would contain intense discussion about mathematics and I already had enough of it. But when I opened and read the first two pages or so of the book, I quickly changed my mind.

          The book opened with the author’s belief that the most interesting topics of mathematics are not taught in school but are usually experienced by the person in the least expected places and that inspired him to make this book. He filled the book with different types of mathematical games, puzzles, trick and even stories that would surely stir the readers mind and interest. I have recognized some of the tricks mentioned for our instructor gave us some of those. The answer to the games and puzzles were given at the latter part of book which makes it very convenient to the reader for they would be able to know if they got the problem right or not. This book unveiled the true beauty of logic, geometry and probability which is associated to different sorts of problem.

          All in all, I found the book quite interesting for the puzzles had truly stirred my interest. The books contained no moral or any of that sort but only delight and amazement. In his quiet way, Stewart has done more than his colleagues have done in the past fast years. The way he wrote the problem was truly amazing and the curiosities were well explained but still remained the problems real tough. This book will certainly challenge both non mathematicians and mathematicians.

An Atypical Tale

              When we were told to read and to make a review about the book: Certain Ambiguity by Gaurav Suri and Hartosh Singh Bal, I immediately turned on my laptop and searched for a PDF for this book for I, a self-declared bookworm, wanted to get my hands on that book and boy I was shocked when the result came out! A mathematical novel? For real? For the past 10 years that I’ve been reading and burying myself with books, not even once I encountered this kind of genre. Who would have thought that some people would even bother to make a mathematical novel? I mean, isn’t it enough that we read them almost every day for an hour or so because they are one of our major subjects?  But still, curiosity killed the cat and unfortunately, that cat was me. I wasn’t really forced to read the book since the outline caught my attention ans made me wonder if the book was really that good. I decided to read it since I believe that it wouldn’t hurt to try and give this book a chance.

          The book opened with a man named Ravi Kapoor who was reminiscing to the time his mathematician grandfather gave him a math problem to try on a calculator. This part indicates the close relationship that the man had with his grandfather. The next day, the grandfather died and which then implies to the reader that Ravi’s grandfather had a big impact on his life for he pursued what he’s grandfather wanted for him to do in the future. But without his grandfather by his side, a man who had a strong passion for mathematics, Ravi became apathetic with the said subject. Still, he strived and got accepted at Stanford University and studied economics due to his father’s urging. There, he met his first friend Peter Cage whose career is inclined with business. As each succeeded page pass by, we learned that Peter and Ravi signed for “Thinking about Infinity” class under Professor Nico who Ravi eventually befriended. Ravi soon found out that his late grandfather Vijay Sahni, was jailed in 1919 at New Jersey and was charged for blasphemy. At this point, there was an alternation of events from Nico’s lecture to Ravi who was searching for the reason behind his grandfather’s imprisonment which then he discovers transcripts of philosophical discussions and a skeptical judge challenging his grandfather to defend his belief. The math topics conferred in the story includes Zeno’s paradoxes, Godel’s Incompleteness, Paul Cohen’s Consistency theorem and most of all the Euclidean Geometry which was the focal point of the story. At the end of the story, we learned that Ravi eventually left economics and chose a profession inclined with mathematics, he also end up with Claire, a fellow student during his infinity class.

          I found the book fascinating and was totally worth it to read. Though I’m not an enthusiast of mathematics, I was able to appreciate the topics discussed in the book for it was new to me and was believable enough. The authors of the book did a great job in explaining faith in a mathematical point of view. All in all, this book was pleasing enough and you can obtain lot of information from it especially in applying basic math axioms in real life situations. If given the chance to read another novel with this kind of genre, I wouldn’t have any second thoughts again J

Another Line of Attack

              When we were task to read the book: Rock, Paper and Scissors by Len Fisher, and give a review about it, I was devastated. My subconscious kept asking why is it again associated with mathematics? Isn’t it enough that we had to live with it ever since we entered kindergarten? But then again, my inquisitiveness overpowered me and so I decided to read the book. Not because we were required to do but for the reason that I am curious and I want to stop this madness.

          In the book, the author Len Fisher generally talked about Game Theory. Game theory, according to Fisher tells us what is going on behind the confrontation, broken promises, and just plain cheating. Aside from the Game Theory he also talked about other complicating factors such Nash’s Trap which implies that the equilibrium always leads to a bad outcome, N-person Situations which is cooperation between two individuals, Perfect and Imperfect Information and lastly, the Simultaneous or Sequential Strategic Decisions.  Fisher also demonstrated the limits of game theory and used it to show how obliging behaviors have evolved to escape the traps created by selfishness and how teamwork persists even in the most challenging situations. The author also suggested employing the same strategy of game theory to solve territorial disputes between countries. He also talked about cooperation and how we can achieve trust, the seven deadly dilemmas which was clearly stated and explained.

              The book was pleasing enough and was fun especially in the end notes part where it is full of stories, jokes and random trivia which will help you in future. Truly, Fisher has a knack for jokes that made the reader that they were reading a book full of witticism. 

Geometry: Rocks, Papers and Toothpicks

 “Sticks and stones may not break my bones, but Geom might somewhat crush me!” Ever since math was introduced to me, geometry was the least of my favorite. I didn’t even know how I survived my high school geometry because solving angles or anything related to it I a brain teaser for me. To be honest, doing a discussion and a group report about geometry really gave me pressure to do good and to prove that geometry, in it hardest concepts, also has an easy side and that through this reporting, I may be able to convince myself that geometry is not just a mere collection of formulas, theorems and postulates, that what we learn from it has applications, too which can be beneficial to human and the society.

When we think of the word “geometry”, the first that we associate directly to it are shapes whether quadrilateral, triangle, pentagon and the like. Having the opportunity to introduce this field to our classmates, we started our first meeting with a form this shape game I called “Search and Form Me.”  Before our classmates entered the room, we already hid figures of different sizes and shapes. Per group, 7 shapes were each assigned to be formed into a bigger shape, the heart (since that was a Valentine’s Week). After all the instructions were given, teams were given an unlimited time to form the unifying shape until one group can form the said shape. It almost took 5 minutes for the first team, the Team Computers, to form the shape. They have been given the necessary points for winning the game and we proceeded to the next game.

The next game that we had is the toothpick game where figure analysis of each player was tested by moving or removing toothpicks to form the shape desired by the game master. Initially, they were given 24 toothpicks and formed it into shapes drawn on the board. The game master would then instruct “Remove 2 toothpicks to form three boxes.” After the go signal, the players then find way to remove toothpicks to form the shape instructed by the game master. Groups who got the right answer were given points. The game was divided into three categories, the easy round (2 points), the average round (3 points) and the difficult round (5 points) where the group having the highest accumulated points from this game plus the first game will win the so-called competition. The players were given 2-3 minutes (sometimes with extension) to finish the task. With this game, you can really see how those imaginative minds work since they were not allowed to move or remove the toothpicks unless they are sure of their answers. We also saw how each group brainstormed and united to beat the challenge. For the first 2 games, Group number Systems led the ranking.

We played the next game on our second meeting which is called the “Form-ulate me” game. Groups were given a mathematical formula corresponding to the formula of the volume, mass, circumference, diameter, or density of a certain shape. After hearing the formula, they have to form the shape through the use of their bodies. Unluckily, we considered the game void due to a problem or misunderstanding experienced by the class. We declared the group Number Systems as the over-all winner and then we presented our hilarious video presentation.

Our video presentation mostly talked about Euclidean Geometry and Non-Euclidean Geometry. We have been able to compare them and explained them through the shapes and lines seen in the buildings of the school. We also talked about the applications of geometry and even incorporated the concept of BBC: The Story of Maths to our video. The video took 8 minutes, full of facts.

Lastly, we had our debate or discussion about “What shape should buildings or any architectural designs have? Should it be a quadrilateral shape or a triangle?” The pros and the cons expressed their thoughts supporting their own stand on the issue. The pros (or the quadrilateral side) said that should be a quadrilateral since most of the buildings were already built that way and that it would save a space more than using a triangular-shaped building. They also claimed that quadrilateral shapes of buildings originated since the early times and mostly followed by a lot of people. The cons counterattacked the other group and pointed out that a triangular building is more stable because of its base and can stand better and stronger. The 2 groups exchanged views until the audience interacted and shared their knowledge about the topic. We had great time absorbing new information about geometry while talking about architecture. The debate ended with a stand that it is on the option of the person involved whether a quadrilateral or a triangular shape is better for buildings.

Overall, I felt great after all the presentations, games and debate were done because I learned something new and something better. It was really fortunate for me to change my views about a certain thing I disliked through my researches. It was also a great reward to hear the applause of other groups plus Ma’am Blasing’s “Job well done” words which gave us a sigh of relief. I realized that geometry really is fun and I would really try hard to discover more about this subject. Though there were some challenges met while we were preparing, all the efforts that we put through paid off, from shooting, video editing, to fact finding. Through rocks, papers and toothpicks, I realized how important geometry is to the lives of the people and how shapes and angles affect each one’s perspective about geometry.

Rocks, Papers, Scissors Book Review

Reading a math book can be a tragedy especially when the one reading it do not have a proper mathematical background. A book concerning math (specifically the game theory), which does not really explains its matters with lots of math, is really an interesting book to read. The Rocks, Papers, Scissors: Game Theory in Everyday Life of Len Fisher links the concepts of game theory to everyday life situation. The book welcomes you to the journey in search for answers to some of our most intriguing social questions.

Have you ever heard of the Prisoner’s Dilemma?  When John Nash discovered the hidden logical trap, it led us to a disturbing series of social dilemmas. In the Tragedy of the Commons, the best overall outcome is produced by cooperation, but self-interest could bring individuals to cheat on the cooperation. An example of what game theorists call the Prisoner’s Dilemma is this story:

There were two thieves caught by the police. The prosecutor has enough evidence to imprison them for two years due to carrying concealed weapon rather than accusing them of burglary with a maximum penalty of ten years. The two thieves plead guilty but the prosecutor has something in mind that could change their pleas. His first encounter was with thieve A. He said that if thieve B plead guilty and he doesn’t, thieve B will only have four years in prison and thieve A will be the one to receive the maximum penalty of ten years. So it is better if thieve A will also plead guilty to receive four years rather than ten. But the prosecutor offered him another deal that if thieve B will not plead guilty and thieve A does, thieve A can be free for turning over state’s evidence. The prosecutor had offered the same talk with thieve B and they both found pleading guilty better than any other choices. As a result, they both got the four-year sentence rather than the two years they might have received if both of them just remained silent.

The world is facing serious problems relating to this Prisoner’s Dilemma. An example is the arms race which activated in the 1950s. The goal is to minimize arms production to use the money for more constructive purposes. This could benefit everyone but no nation could afford to be disarmed so long as other countries continued to build up stocks of nuclear weapons. The global warming which is a menace to everyone is still not given the right act ion because it is still troubled by a contradicting logic. Many polluting countries do not see the benefit of controlling carbon emissions because there are also other nations which cause continuous pollution.

Social dilemmas were at last understood through the discovery of John Nash in 1949. He made his discovery of basic logical trap when he was just twenty-one and not yet having his mental illness, schizophrenia. Nash was a genius. He was just eighteen when he identified the logical trap now called as the Nash Equilibrium: “It is a position in which both sides have selected a strategy without ending up in a less desirable position.”

According to the author, communication and negotiation are the twin keys that can unlock social dilemmas. Through these keys, we are permitted to share information and our own ideas to others. In this way alliances can be formed wherein coordination and joint agreement can coexist. Also, according to him, effects of social dilemmas of game theory and the real world concerns trust issues. If we will be able to trust one another, dilemmas could suddenly disappear. When we trust each other, we are free to deliver our thoughts to others. We can negotiate about things which needed our actions and we can coordinate our strategies to formulate best solutions. We should not break agreements just because of our selfish acts that patronize personal interests.  We should be able to trust and be able to free our minds with the possibilities of being cheated. Free yourself from the act of retaliating. As what Oscar Wilde said, “always forgive your enemies, nothing annoys them so much.”

An Individual Report for Algebra

                On the first meeting of MST3 section B, the whole class was divided into 5 groups. Also, there were five topics in this course that would be taken up namely Number Systems, Algebra, Geometry, Statistics and Computer. We were asked by our teacher to select one among the five topics and we were tasked to have a debate and make a creative presentation and a game about that. Our group selected Algebra and we were scheduled to have our presentation and debate on the first week of February.

                First, I thought that presenting Algebra in a creative way would be easy and also having a debate about it won’t be so hard but when the week were we would perform came,  we started to panic because we didn’t know what to do and by that time, we were busy for our other major subjects.

                We didn’t have much time to prepare for our presentation because we haven’t planned anything so we decided to have our game first and we had a quiz bowl show. The whole quiz bowl duration, I was so nervous because there was some misunderstanding with the rules and regulation of our game but eventually, it went fine. Then there was a game where a puzzle was scattered at the school grounds then again, there was a little problem because a puzzle set was not done for the Computer group but one of my group mate decided to add points to all of the groups so that it wouldn’t be unfair to the Computer group. The next meeting, we had our presentation, it was impromptu and I was so nervous and ashamed because we had a live performance. It was about a girl who had problems in Algebra and her friends encouraged her by sharing what were the history and other important uses of Algebra in the society. In the live performance, I was the one who acted as the problematic girl and I was ashamed because I really don’t do something like that. After the life performance, we had our debate. It was about whether Algebra is essential to our everyday life or not. Of course, it was essential. You are not aware but you are already doing algebra everyday because it is an organized arithmetic.

                Presenting algebra, having games and debate about it is not that easy especially when your group mates came from a different bloc and even from a different course. It was hard because you have no same free time and it was in the midst of many examinations. But at least, we still have done our task and we are thankful that were not reprimanded by our professor. I have learned one thing in what we have done. When there is something to do, you should not do it the night before or even minutes before it is going to be submitted or performed. 

A Book, A Novel, A Fiction

From kindergarten to high school – and even in college, we learned mathematics through reading math books and textbooks. Line by line, we were to read the lessons in the hope that we learn the concepts and principles behind those mathematical ideas, axioms and theorems. Line by line we read. Line by line we studied. And line by line we memorized. But let us face it, we don’t remember most of what we read, studied and memorized. The lines of those mathematical books have been left in the past and have long been forgotten.

We may have read them, studied them and memorized them but we have failed to understand them. And if given a mathematical textbook, most students won’t even flip the page if their teachers won’t force them to. And this is most probably because we easily get bored reading mathematical books because it is written in way that is so formal, so straightforward and just simply so boring. To put things simply, we believe mathematics is boring.

Contrary to this belief, the book A Certain Ambiguity: A Mathematical Novel written by Indian authors Gaurav Suri and Hartosh Singh Bal shows us that reading mathematics too can be fun and fascinating.  A winner of the 2007 Award for Best Professional/Scholarly Book in Mathematics, this mathematical fiction talks about finding certainty in mathematics and philosophy. (Wikipedia, 2013)

This compelling novel talks about the journey of Ravi Kapoor, of Indian nationality who pursued his studies in Stanford University, America, where he discovers that he confronts the same mathematical and philosophical dilemmas that his mathematician grandfather had faced before – which had been a factor that landed him in jail. The grandfather, Vijay Sahni, was charged in 1919 under an obscure blasphemy law in a small town in New Jersey. The skeptical judge challenged him to defend his belief that the certainty of mathematics can be extended to all human knowledge – even religion. (press.princeton.edu, 2013)

In the novel, Kapoor and his grandfather struggled with the questions of “whether there can ever be absolute certainty in mathematics or life?” Along with contemplating these very philosophical question is the act of reconsidering their fundamental beliefs and perspectives. Their journey in queering these mathematical questions led them to explore the parallel development that exists in the study of geometry and infinity. (press.princeton.edu, 2013)

Beginning with a nostalgic flashback experienced by Ravi Kapoor to the time his mathematician grandfather gave him a math problem to try on a calculator, the story starts with the passing along of the grandfather’s interest on mathematics to his grandson Ravi. Upon Vijay’s death, Ravi inherits a large sum of money for the purpose of sending him to college in America. And soon Ravi gets accepted to Stanford where he majors in Economics. (Vestal, D., 2007)

During his senior year, Ravi decides to take a class called “Thinking About Infinity”, and in this class he learns more than what he expected to learn. (Vestal, D., 2007)

In his class, not only did he learn about mathematical principles and concepts on infinity but as well as the facts about his grandfather spending some time in America which landed him in jail for blasphemy. Wanting to dig deeper into it, he reads court documents to find out what exactly happened. From there he finds out that the American judge, John Taylor, who was trying to decide what to do with his grandfather was a Christian who looks to religion for certainty while his grandfather, who was an atheist, looks to mathematics for certainty. However Vijay’s faith in mathematics was shaken by the appearance of non-Euclidean geometry in astrophysics.  (Vestal, D., 2007)

The novel tells us a lot of mathematical concepts and philosophy mostly through the author’s act of incorporating them to the lectures that Ravi attends. (Vestal, D., 2007) Some of these math topics that were discussed in the book were elementary mathematics of infinity, set theory, Euclidean geometry, non-Euclidean geometry, Zeno’s paradoxes and infinitude of primes through Godel’s incompleteness and Paul Cohen’s Consistency Theorems. (Bogomolny, A., n.d.) The authors also presented the history of mathematical certainty, and they have done this through imagining things like letters and journal entries from Pythagoras, Nicole Oresme, Cantor and many more. (Vestal, D., 2007) And while doing this, they have presented mathematics as a human endeavor.  (Bogomolny, A., n.d.)  (The authors have stressed that these writings were purely fictional and have expounded the history in the notes section by the end of the book). (Vestal, D., 2007)

The book ends with Ravi choosing the career leading to mathematics instead of the path leading to economic (which could have led him to make more money). (Bogomolny, A., n.d.) 

This novel is unlike any other math books I have encountered. Unlike the usual math books and textbooks, this book is compelling in every aspect you look at it. It is fun and fascinating without jeopardizing its mathematical content.  This is unlike any other math books. This is a mathematical novel. This is a mathematical fiction. And most of all this is a mathematical book.
 
Mathematical Novel. Math books are usually written in a manner that is formal, straightforward and full of jargons. They may be understandable to those within the academe of mathematics but for those like me, we see a jumble of words which makes it hard for us to understand the concepts behind them. However, this book has shifted anew, grasping another style. From being a boring math textbook, it wanted to metamorph into something different, something that people would actually want to take from the shelf, something that people would actually want to flip the pages, and something that people would actually want to look forward to read during their leisure and free time. It became something. It became a novel. And we have to remember that novel means there is a story behind those pages, and people love stories. People don’t like reading explanations and boring definitions, they want to feel the thrill, the mystery and the excitement that the characters feel in the novel. And with this book being a novel instead of the usual textbook, people are going to be enticed to read it. And instead of just forgetting the concepts people read in mathematical textbooks, in a novel, we would remember it more because they can associate it to circumstances and stories within the novel.

Mathematical Fiction. Aside from being a novel, this book is under the genre, fiction. That it is make-believe. And this ignites desire from people to actually read this book. People are more inclined to read fiction because men are adept to imagining things. Aside from this, people want things to happen, that is why more often than not, we find ourselves day dreaming of a better life and of a better status. And aside from this, fictional stories allow us to think ourselves. It stretches our imagination. We want to know why it happened, what could happen and what could have happened? It even ignites the curiosity that is within us. And if we are curious we actually dig deep into things. This is what happens once you read this book, you get curious and you end up searching and digging deeper into things. And we actually get more attached to it. We might even start looking up on mathematical concepts that were mentioned in the book just to feed our curious minds. And thus we learn more from this thought provoking book.

Mathematical Book. But above all, this is a mathematical book. Mathematical ideas, concepts and principles were intertwined with the plot and the characters of the story to make it more interesting. From the start of the book, we have come across different mathematical topics already. Almost every page of the book is talking about mathematical concepts, ideas and principles. We encounter mathematics in the different scenes and situation in the story. We learn new concepts, theorems and principles through Ravi’s experience, intellect and analysis. The plot, the character, the genre, is all but a cover to the true essence of the book, which is to bring mathematics to its readers. And it has done it so successfully. What is meant here is that, this book is so exciting that we actually read and understand mathematical concepts, ideas and principles better because we are inclined to the excitement of what is going on in the story. Let us face it, people can’t finish one whole mathematical textbook and understand rather, we’ll just confuse ourselves with the jargons and confusing mathematical axioms. We might not even finish one whole chapter. But in this book we not only have a novel and a fiction but what is most important is we learn and understand the mathematical concepts and topics that the authors want to impart on us. We actually get acquainted with more mathematical concepts and topics as we read deeper into the story all because many mathematical topics are intertwined with the flow of the story.

This book has provided us with new perspective into looking at mathematics. Who would have thought that mathematics could be intertwined with novel? Who would have thought that mathematics could be this interesting? But most of all whoa would have thought that the beauty of mathematics could be reflected through this book being a novel, a fiction and a mathematical book.
Before when we use to read math textbook, line by line we read, line by line we study, and line by line we memorize, yet through those lines we failed to understand and remember.

Now, though this book being a novel and through the fictional story, we have come to appreciate math and its beauty.



REFERENCE:

Bogomolny, A. (n.d.) Reviewed by Alexander Bogomolny [Review of the book  A Certain Ambiguity: A Mathematical Novel]. Cut the Knot. Retrieved February 13, 2014 from http://www.cut-the-knot.org/books/Reviews/CertainAmbiguity.shtml

Vestal, D. (2007, September 26). Reviewed by Donald L. Vestal, on 09/26/2007 [Review of the book A Certain Ambiguity: A Mathematical Novel]. Mathematical Association of America. Retrieved February 13, 2014 from http://www.maa.org/publications/maa-reviews/a-certain-ambiguity-a-mathematical-novel

A Certain Ambiguity. (2013, December 12). In Wikipedia. Retrieved February 13, 2014 from http://en.wikipedia.org/wiki/A_Certain_Ambiguity

A Certain Ambiguity: A Mathematical Novel. (2014, January 14) In Princeton University Press. Retrieved February 13, 2014 from http://press.princeton.edu/titles/8479.html