Game Theory - Interesting notes

Finishing a very interesting Audio course on 'Game Theory', which was a field that really interested because of its relationship with strategy, probably, mathematics and management science, I couldn't but notice some points that truly describe the elegance of sciences and knowledge through their beautiful interconnectivity.

Here are some of my observations and highlights:

• Game Theory can be defined as the 'Science of strategic decision making in an interactive environment'. A strategy in the game theory context is the best move or choice that a player makes in every situation to maximize his payoff.

• Beautifully enough, rationality is defined in game theory as 'seeking the solution that maximizes the payoff according to a certain player's standards'.. Makes you think about how much some people's lives would improve if they just were rational enough to define their standards and work according to them. Having the 'right' standards is a whole new issue by the way.

• The nice concept of a 'Nash' equilibrium, which is defined as a scenario in which any player would be 'satisfied' by his strategy so that if he could, he wouldn't change

• This is a nice a thought exercise called 'the prisoner's dilemma'..
  
  Two prisoners are with the police and being investigated as police doesnt have enough evidence. They are both guilty. They are separated and each told that if he 'tells' he will be rewarded by being set free, the other will receive the full penalty. If none 'tells' they both receive minimal prison time; yet if they both tell, they will share the full penalty. This is a non-zero sum game, and even though any rational player will chose to 'tell' on the other player, the outcome of this is much worse than cooperation... No Nash equilibrium exists here... if a player 'tells'; and if the other doesn't then its fine, but then that other would wish if he had made the other choice.. If the other had told, then both would wish that they hadn't told... Etc..

• A study of game theory, combined with statistics of penalty kicks shot sides (in soccer) and goal keeper jumps would lead to the conclusion (according to the studied sample) that best strategy for a goal keeper is to go left with a 42% probably thus reducing the shooter's success chance to its lowest @ 83%.... Surprisingly enough the shooter's best strategy would be to shoot left with a 39% probability, thus reaching the success chance of 82%.. this is a minimax game strategy.

• This is another interesting thought exercise.. Its called the Monty Hall problem (after the show) - It was actually in the movie '21' but - naturally - they failed to explain it clearly, and it happened that a friend asked me about it, Here is a summary and explanation:
  
  There is a car behind 1 of 3 doors, a player selects one door at random, so the host then opens one of the other two doors, the one which he knows surely doesn't have the car behind it. The player then has two options: Either keep that same choice, or chose another. WHAT SHOULD HE DO??
  
  Here initially the player had a 33% chance of winning (1 out 3).
  
  After the opening of one 'wrong' door, game theory proves that the best strategy is to CHANGE the door - ALWAYS.. Why??
  
  Well, If the player got it right initially ( something of 33% probability) and changed then he will always lose - this has a 33% chance of happening ... If he had got it wrong (66% chance) and changes he will always win. The total payoff is the sum of weighted payoffs, which will be 66% when changes, 33% when he doesn't .. ALWAYS... Experimentally this is true as well, as counter intuitive as it may seem.

• This is interesting for computer scientists.. One experiment asked programmers to come up with a computer program that can compete with people in the 'prisoner's dilemma' game, and there were many programs submitted that we actually tested with a large number of subjects. There was one program that - IN PRACTICE - always did best:
  the tit-for-tat, that is, if you betray me this round, i will betray you in the next one, otherwise i will cooperate......... Even though there were many formulas that if you calculate will prove to be better theoretically, this program did far better than them (after all this program's best possible result is a tie)..... Why? well it has the magical components: It is simple, provok-able, forgiving and straight forward... In essence it is positive, simple and just.... This is a deep insight about humanity!

• Game theory analyzes threats, promises, commitments, signalling and incentive schemes in depth, along with the elements that make them work and useful

• An interesting issue is the 'tragedy of the commons' defined as the scenario in which a group of people when doing a certain 'bad' activity in a number that exceeds a certain threshold, will all suffer damages and losses... A small number of them doing it will not have that effect.. this originated from sharing grazing lands, but has very nice applications to pollution and traffic jams for example.

• Applying game theory to economics in determination of demand equations and optimum profit levels (Oligapolies)

• Applying game theory to voting methodologies and determining the effects of each style of voting and democracy for example on overall fairness (the tyranny of the majority, the average preference, etc..)

• The application of game theory to Auctions and their types, and the methods of generating maximum return to the auctioneer and the maximum value of the bid-winner, introducing the beautiful dilemma of the 'winner's curse' ( you only win if you pay for something much more than anybody else thinks its worth )

This was a really challenging course that can raise a lot of healthy scientific curiosity to the underlying mechanism of many behaviors that sometimes 'just happen' , and can help a lot in supplying us with the tools and methodologies to have a good grasp and analysis of gain and strategy...