# Game Theory_ 2.pptx

4. Nov 2022
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### Game Theory_ 2.pptx

• 1. Game Theory
• 2. Application of game theory Game theory is the process of modeling the strategic interaction between two or more players in a situation containing set rules and outcomes. While used in a number of disciplines, game theory is most notably used as a tool within the study of economics. The economic application of game theory can be a valuable tool to aide in the fundamental analysis of industries, sectors and any strategic interaction between two or more firms.
• 3. Game Theory Definitions 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.
• 4. Game Theory Definitions • Game: Any set of circumstances that has a result dependent on the actions of two of more decision-makers (players) • Players: A strategic decision-maker within the context of the game • Strategy: A complete plan of action a player will take given the set of circumstances that might arise within the game • Payoff: The payout a player receives from arriving at a particular outcome. The payout can be in any quantifiable form, from dollars to utility. • Information set: The information available at a given point in the game. The term information set is most usually applied when the game has a sequential component. • Equilibrium: The point in a game where both players have made their decisions and an outcome is reached.
• 7. Key Elements of a Game
• 8. Assumptions Used in Game Theory • 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. • When examining games that are already set up, it is assumed on your behalf that the payouts listed include the sum of all payoffs associated with that outcome. 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. One of the simplest games is a sequential game involving two players.
• 9. Solving Sequential Games Using Backwards Induction Below is a simple sequential game between two players. The labels with Player 1 and Player 2 within them are the information sets for players one or two, respectively. 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).
• 10. 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.
• 11. 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". Below is the solution to the game with the equilibrium path bolded.
• 12. Prisoner’s Dilemma The prisoner’s dilemma scenario works as follows: Two suspects have been apprehended for a crime and are now in separate rooms in a police station, with no means of communicating with each other. The prosecutor has separately told them the following: • If you confess and agree to testify against the other suspect, who does not confess, the charges against you will be dropped and you will go scot-free. • If you do not confess but the other suspect does, you will be convicted and the prosecution will seek the maximum sentence of three years. • If both of you confess, you will both be sentenced to two years in prison. • If neither of you confess, you will both be charged with misdemeanors and will be sentenced to one year in prison.
• 13. Zero-Sum Game Zero-sum is a situation in game theory in which one person’s gain is equivalent to another’s loss, so the net change in wealth or benefit is zero. A zero-sum game may have as few as two players, or millions of participants. Zero-sum games are found in game theory, but are less common than non-zero sum games. Poker and gambling are popular examples of zero-sum games since the sum of the amounts won by some players equals the combined losses of the others. Games like chess and tennis, where there is one winner and one loser, are also zero-sum games. In the financial markets, options and futures are examples of zero-sum games, excluding transaction costs. For every person who gains on a contract, there is a counter-party who loses.
• 14. Example One player’s gain is the other’s loss. The payoffs for Players A and B are shown in the table below, with the first numeral in cells (a) through (d) representing Player A’s payoff, and the second numeral Player B’s playoff. As can be seen, the combined playoff for A and B in all four cells is zero.
• 15. Zero-sum games are the opposite of win- win situations – such as a trade agreement that significantly increases trade between two nations – or lose-lose situations, like war for instance. In real life, however, things are not always so clear-cut, and gains and losses are often difficult to quantify.
• 19. The dominant strategy A strategy is dominant if, regardless of what any other players do, the strategy earns a player a larger payoff than any other. Hence, a strategy is dominant if it is always better than any other strategy, for any profile of other players' actions. Depending on whether "better" is defined with weak or strict inequalities, the strategy is termed strictly dominant or weakly dominant. If one strategy is dominant, than all others are dominated. For example, in the prisoner's dilemma, each player has a dominant strategy.
• 20. Strictly dominant strategy A strategy is strictly dominant if, regardless of what any other players do, the strategy earns a player a strictly higher payoff than any other. Hence, a strategy is strictly dominant if it is always strictly better than any other strategy, for any profile of other players' actions. If a player has a strictly dominant strategy, than he or she will always play it in equilibrium. Also, if one strategy is strictly dominant, than all others are dominated.
• 21. Weakly Dominant Strategy A strategy is weakly dominant if, regardless of what any other players do, the strategy earns a player a payoff at least as high as any other strategy, and, the strategy earns a strictly higher payoff for some profile of other players' strategies. Hence, a strategy is weakly dominant if it is always at least as good as any other strategy, for any profile of other players' actions, and is strictly better for some profile of others' strategies. If a player has a weakly dominant strategy, than all others are weakly dominated. If a strategy is always strictly better than all others for all profiles of other players' strategies, than it is strictly dominant.
• 22. Dominated Strategies Are there obvious predictions about how a game should be played?
• 23. A Nash equilibrium, named after John Nash, is a set of strategies, one for each player, such that no player has incentive to unilaterally change her action. Players are in equilibrium if a change in strategies by any one of them would lead that player to earn less than if she remained with her current strategy. For games in which players randomize (mixed strategies), the expected or average payoff must be at least as large as that obtainable by any other strategy. Nash Equilibrium
• 24. Mixed Strategy A strategy consisting of possible moves and a probability distribution (collection of weights) which corresponds to how frequently each move is to be played. A player would only use a mixed strategy when she is indifferent between several pure strategies, and when keeping the opponent guessing is desirable - that is, when the opponent can benefit from knowing the next move.
• 25. Minimax Strategy In Minimax the two players are called maximizer and minimizer. The maximizer tries to get the highest score possible while the minimizer tries to do the opposite and get the lowest score possible. Minimax is a decision rule used to minimize the worst-case potential loss; in other words, a player considers all of the best opponent responses to his strategies, and selects the strategy such that the opponent's best strategy gives a payoff as large as possible. The name "minimax" comes from minimizing the loss involved when the opponent selects the strategy that gives maximum loss, and is useful in analyzing the first player's decisions both when the players move sequentially and when the players move simultaneously. In the latter case, minimax may give a Nash equilibrium of the game if some additional conditions hold.
• 26. Type of equipment Modification М-1 М-2 М-3 А-1 10 6 5 А-2 8 7 9 А-3 7 5 8 Example Upper price of the game
• 27. Type of equipment Modification М-1 М-2 М-3 А-1 10 6 5 А-2 8 7 9 А-3 7 5 8 Example Lower price of the game