There are also other ways to control how we randomize. For example, for stone-paper-scissors we can toss a six-sided die and decide to select stone half the time the numbers 1, 2 or 3 are tossed , select paper one third of the time the numbers 4 or 5 are tossed or select scissors one sixth of the time the number 6 is tossed. Doing so would tend to hide your choice from your opponent. But, by mixing strategies in this manner, should you expect to win or lose in the long run? What is the optimal mix of strategies you should play? How much would you expect to win?
This is where the modern mathematical theory of games comes into play. Games such as heads-tails and stone-paper-scissors are called two-person zero-sum games. Zero-sum means that any money Player 1 wins or loses is exactly the same amount of money that Player 2 loses or wins.
That is, no money is created or lost by playing the game. Most parlor games are many-person zero-sum games but if you are playing poker in a gambling hall, with the hall taking a certain percentage of the pot to cover its overhead, the game is not zero-sum. For two-person zero-sum games, the 20th centurys most famous mathematician, John von Neumann, proved that all such games have optimal strategies for both players, with an associated expected value of the game. Here the optimal strategy, given that the game is being played many times, is a specialized random mix of the individual pure strategies.
The value of the game, denoted by v, is the value that a player, say Player 1, is guaranteed to at least win if he sticks to the designated optimal mix of strategies no matter what mix of strategies Player 2 uses. Similarly, Player 2 is guaranteed not to lose more than v if he sticks to the designated optimal mix of strategies no matter what mix of strategies Player 1 uses. If v is a positive amount, then Player 1 can expect to win that amount, averaged out over many plays, and Player 2 can expect to lose that amount.
The opposite is the case if v is a negative amount. That is, both players can expect to win 0 over a long run of plays. The mathematical description of a zero-sum two-person game is not difficult to construct, and determining the optimal strategies and the value of the game is computationally straightforward.
We can show that heads-tails is a fair game and that both players have the same optimal mix of strategies that randomizes the selection of heads or tails 50 percent of the time for each. Stone-paper-scissors is also a fair game and both players have optimal strategies that employ each choice one third of the time. Not all zero-sum games are fair, although most two-person zero-sum parlor games are fair games. So why do we then play them?
Cooperative game theory
They are fun, we like the competition, and, since we usually play for a short period of time, the average winnings could be different than 0. The Skin Game: Two players are each provided with an ace of diamonds and an ace of clubs.
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Player 1 is also given the two of diamonds and Player 2 the two of clubs. In a play of the game, Player 1 shows one card, and Player 2, ignorant of Player 1s choice, shows one card. Player 1 wins if the suits match, and Player 2 wins if they do not. The amount payoff that is won is the numerical value of the card of the winner. But, if the two deuces are shown, the payoff is zero. This game is a carnival hustlers Player 1 favorite; his optimal mixed strategy is to never play the ace of diamonds, play the ace of clubs 60 percent of the time, and the two of diamonds 40 percent of the time.
We can have many-person competitive situations in which the players can form coalitions and cooperate against the other players; many-person games that are nonzero-sum; games with an infinite number of strategies; and two-person nonzero sum games, to name a few. Mathematical analysis of such games has led to a generalization of von Neumanns optimal solution result for two-person zero-sum games called an equilibrium solution. An equilibrium solution is a set of mixed strategies, one for each player, such that each player has no reason to deviate from that strategy, assuming all the other players stick to their equilibrium strategy.
We then have the important generalization of a solution for game theory: Any many-person non-cooperative finite strategy game has at least one equilibrium solution.
By now you have concluded that the answer to the opening question on competitive situations is "game theory. Any time we have a situation with two or more players that involves known payouts or quantifiable consequences, we can use game theory to help determine the most likely outcomes. As with any concept in economics, there is the assumption of rationality. There is also an assumption of maximization. It is assumed that players within the game are rational and will strive to maximize their payoffs in the game. This will exclude any "what if" questions that may arise.
The number of players in a game can theoretically be infinite, but most games will be put into the context of two players.
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One of the simplest games is a sequential game involving two players. Below is a simple sequential game between two players. The numbers in the parentheses at the bottom of the tree are the payoffs at each respective point. The game is also sequential, so Player 1 makes the first decision left or right and Player 2 makes its decision after Player 1 up or down.
Backwards induction, like all game theory, uses the assumptions of rationality and maximization, meaning that Player 2 will maximize his payoff in any given situation. At either information set we have two choices, four in all. By eliminating the choices that Player 2 will not choose, we can narrow down our tree. In this way, we will bold the lines that maximize the player's payoff at the given information set.
After this reduction, Player 1 can maximize its payoffs now that Player 2's choices are made known. The result is an equilibrium found by backwards induction of Player 1 choosing "right" and Player 2 choosing "up. For example, one could easily set up a game similar to the one above using companies as the players. This game could include product release scenarios.
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If Company 1 wanted to release a product, what might Company 2 do in response? Will Company 2 release a similar competing product? By forecasting sales of this new product in different scenarios, we can set up a game to predict how events might unfold. Below is an example of how one might model such a game. By using simple methods of game theory, we can solve for what would be a confusing array of outcomes in a real-world situation. Using game theory as a tool for financial analysis can be very helpful in sorting out potentially messy real-world situations, from mergers to product releases.
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