Matrix two player game

Since we have assumed that our column player acts rationally, we can expect them to act in accordance with the stochastic vector x. In other words, the column player has adopted strategy x.

The above equation can be solved by reformulating it as a linear program. By taking the inner optimization over the deterministic strategies, this equation can be re-written as:. In order to put a lower bound on the minimization term, a new variable v is introduced. This gives us the following linear program:. The Minimax Theorem can now be used to verify that both solutions are consistent with one another. The value of a game shows how much utility each player can expect to gain or lose on average.

In order to illustrate the power of the minimax theorem in solving matrix games, a numerical example has been provided in the section below.

Many decisions made in sports can be modeled as finite two-person zero-sum games. Take, for example, a common dilemma seen in American football. The offense has driven down the field and is just a few short yards of scoring. The team has four plays, or downs , to score. On the third down, the team gets stopped by the defense and is unable to score, leaving only one more play to make it happen. There are two options for scoring. The first is a field goal, in which the team kicks the ball through the uprights for 3 points.

The second option is to run a passing or running play for a touchdown, worth 7 points. While the option of scoring a touchdown yields a higher payoff, it is a much risker option as running and passing plays are easier to defend against than a field goal.

For this reason, football coaches often settle on kicking a field goal on 4 th down instead of going for it. This anticlimactic end to a long and exciting drive often leaves fans with an unsatisfying feeling, knowing that their team was only a few yards from scoring a touchdown. While kicking the field goal nearly guarantees 3 points, is it smarter to employ a more aggressive strategy and go for the touchdown? Game theory can help determine the strategy that will yield the highest amount of points on average over time.

There are a few assumptions to be made in order to model this Fourth and Goal Dilemma. The first is that both football teams are ideal. What this means is that if the offense chooses a run play and the defense chooses to defend a run play, then the run will be stopped with zero yards gained.

Asked 8 years ago. Active 8 years ago. Viewed 1k times. Improve this question. N 2, 10 10 gold badges 32 32 silver badges 56 56 bronze badges. How is this question different from your last question about this? I think you need to re-read the comments there about how to solve the problem break things up into small pieces, test each piece. I made it smaller. N: To improve a question, edit the question, don't repost it. There's an "edit" link near the bottom.

But I think the other suggestions you were given are going to be more helpful than trying to shoehorn this question into SO's format. N I spent some time trying to understand the rules of the game, but still clueless. Could you please give step-by-step operations performed by each player on original matrix? Or give better explanations to the game rules?

Show 5 more comments. You must shoot the waves of Sentinels before they destroy your APUs. A downloadable virtual model ship builder you could download in PC or Mac versions. This game was also available on the official Matrix website in the arcade section. All of the games offered in the arcade section were available free of charge. Matrix Wiki Explore. The Animatrix Animatrix 2.

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