How Many Pure Strategies Are Available To Each Player?

Is every Nash equilibrium Subgame perfect?

Definition 11.1 A Nash equilibrium is said to be subgame perfect if an only if it is a Nash equilibrium in every subgame of the game.

The third subgame is the game itself.

Note that, in each subgame, the equilibrium computed via backward induction remains to be an equilibrium of the subgame..

How do you find the Subgame perfect equilibrium?

To solve this game, first find the Nash Equilibria by mutual best response of Subgame 1. Then use backwards induction and plug in (A,X) → (3,4) so that (3,4) become the payoffs for Subgame 2. The dashed line indicates that player 2 does not know whether player 1 will play A or B in a simultaneous game.

How do you calculate number of strategies?

Therefore the number of possible pure strategies is equal to the number of ways you can pick an action from information set 1 times the number of ways you can pick an action from information set 2, etcetera, up to information set N. In otherwords, it is equal to ∏Nn=1Mn.

How many information sets does each player have?

Both players have 3 information sets. 2. For both players each information set has 2 actions.

What is normal form in game theory?

In game theory, normal form is a description of a game. … The normal-form representation of a game includes all perceptible and conceivable strategies, and their corresponding payoffs, for each player.

What is the difference between pure strategy and mixed strategy?

If a player randomly chooses a pure strategy, we say that the player is using a “mixed strategy.” In a pure strategy a player chooses an action for sure, whereas in a mixed strategy, he chooses a probability distribution over the set of actions available to him.

How do you prove Nash equilibrium?

If each player has chosen a strategy—an action plan choosing its own action based on what it has seen happen so far in the game—and no player can increase its own expected payoff by changing its strategy while the other players keep theirs unchanged, then the current set of strategy choices constitutes a Nash …

Does Nash equilibrium always exist?

There does not always exist a pure Nash equilibrium.

What are pure strategies?

A pure strategy provides a complete definition of how a player will play a game. In particular, it determines the move a player will make for any situation they could face. A player’s strategy set is the set of pure strategies available to that player.

What is a Subgame in game theory?

In game theory, a subgame is any part (a subset) of a game that meets the following criteria (the following terms allude to a game described in extensive form): It has a single initial node that is the only member of that node’s information set (i.e. the initial node is in a singleton information set).

Can you have 2 Nash equilibrium?

Overall, an individual can receive no incremental benefit from changing actions, assuming other players remain constant in their strategies. A game may have multiple Nash equilibria or none at all.

How do you do backward induction?

Backward induction starts to solve the game from the final nodes. Player 2 will make 4 comparisons in total. He will choose an option with the higher payoff. For example, considering the first subgame, payoff of 11 is higher than 7.

What is difference between normal form game and extensive form game?

In normal form games, the matrix demonstrates the strategies adopted by the different players of the game and their possible outcomes. On the other hand, extensive form games are the one in which the description of game is done in the form of a decision tree.

How many strategies does Player 1 have?

3 strategiesA strategy is a complete contingent plan, saying what the player will do at every “point” (or “node”). So in this simple example, player 1 only has 3 strategies, each corresponding to her action. However, note that player 2 gets to observe player 1 before responding.

What is pure strategy Nash equilibrium?

In plain terms, a pure Nash equilibrium is a strategy profile in which no player would benefit by deviating, given that all other players don’t deviate. Some games have multiple pure Nash equilib ria and some games do not have any pure Nash equilibria.

What is extensive form in game theory?

An extensive-form game is a specification of a game in game theory, allowing (as the name suggests) for the explicit representation of a number of key aspects, like the sequencing of players’ possible moves, their choices at every decision point, the (possibly imperfect) information each player has about the other …

How do you solve for Nash equilibrium?

To find the Nash equilibria, we examine each action profile in turn. Neither player can increase her payoff by choosing an action different from her current one. Thus this action profile is a Nash equilibrium. By choosing A rather than I, player 1 obtains a payoff of 1 rather than 0, given player 2’s action.

What is a proper Subgame?

The part of the game tree consisting of all nodes that can be reached from x is called a subgame. … A subgame on a strictly smaller set of nodes is called a proper subgame. A subgame perfect equilibrium is a strategy profile that induces a Nash equilibrium in each subgame.

How many Subgames are there in this game?

6a) There are 6 proper subgames in this game.

What is strict dominance?

-a strictly dominant strategy is that strategy that always provides greater utility to a the player, no matter what the other player’s strategy is; -a weakly dominant strategy is that strategy that provides at least the same utility for all the other player’s strategies, and strictly greater for some strategy.

What is a strictly dominated strategy in a normal form game?

A strategy is strictly dominated if choosing it always gives a worse outcome than choosing an alternative strategy, regardless of which moves other players make. A strategy is weakly dominated if choosing it always gives an outcome that is as good as or worse than choosing an alternative strategy.