Sunday, 23 April 2017

Games Theory

Games Theory

Games Theory

                                                             Game theory is to determine the rules  of rational behavior in game .A game is finite when each player has a finite number  of moves and finite number of choices at each move.It provides a systematic quantitative method for analyzing  competitive situations in which the competitors  make use of logical process and techniques in order to determine an optimal strategy for winning .It is  depending on such factors as the number  of players ,sum of gains and losses and number of strategy employed in the game .For example,if the number of players are two,we refer to the game as two  person game .
game theory ruly

                                                                The zero -sum property  means  that payoff to one player and payoff to other  players together sum to zero.This means  that one player's gain is another's  loss and sum of net gain is zero.                                    


Some rules of the game are                                                                                        

1. The players act rationally and intelligently.                                      2. Each player has available to him to a finite set of possible courses of action.               
3. The players attempt to maximize gain and minimize losses.           4. All the relevant information is known to each player.                     5. The players make individual decisions without direct communication.                                
6. The players simultaneously select their respective course of action.                                                                   
7. The payoff is fixed and known in advance.
         

Sunday, 2 April 2017

Logit model example


The Logit  Transformation   

The Logit  Transformation

                                                                                                                 The logit is the log-odds of the probability. We have seen the idea of a fitting straight line but have not encountered a constraint on the fitted models, such as p always remaining between 0 and 1, as in this case. The solution is to introduce a new class of models. We define the logit of p as the logarithm(base e) of the odds.Mathematically, logit (p)= log(p/(1-p)). The ratio p/(1-p) is the odds or ratio of the probability of the event to the probability of the complementary event. The odds ratio is familiar to epidemiologists and horse-race handicappers alike. The logist is the log-odds of an event occurring. In epidemiology, we usually talk about the odds rather than the probability of an event , especially when the events are rare.                                                      It does not matter much which event is considered a "success" or" failure" in the binomial distribution. Recall the property of the logarithm of a reciprocal for any positive number z.   log(1/z)= -log(z)                                                                                               so we have logit (p)= log(p/(1-p))= -logit(1-p)               


This is, the logit of 'success' is the negative of the logit of "failure". We only need to remember which outcome we are calling a "success". The logit is the log-odds of the probability.                                                                                                                                                  Let us take a moment to motivate to use of the logit. The p parameter is restricted to values between 0 and 1. The logit transforms p to cover the entire number line.A plot of logit (p) against p is given in the figure.From this figure we see how values o p are spread out from the interval of 0 to 1, this transformation takes of extremely large negative and positive values, respectively.                                                                            From this figure we can see that logit (p) is 0 when p is 1/2.similarly, logit(p) is negative when p is less than 1/2. The plot of logit(1-p) in this figure demonstrates the relation given at equation, the logit of "failure" is the negative  of the logit of "success"
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