Ordinal Strategic Games: Ranking Outcomes In Strategy

Ordinal strategic games with equivalent preferences represent a class of strategic games in which players have preferences over outcomes that are merely ordinal, meaning that they can only rank the outcomes but cannot assign numerical values to them. These games are characterized by the absence of interpersonal comparisons of utility, the assumption that players have complete and transitive preferences, and the presence of a common set of outcomes. Most notable examples of ordinal strategic games with equivalent preferences include zero-sum games, coordination games, and games with externalities.

Structure for Ordinal Strategic Games with Equivalent Preferences

In ordinal strategic games, players have preferences over outcomes that can be ranked but not measured in terms of absolute differences. When players have equivalent preferences, meaning they all rank the outcomes in the same way, the social welfare function used to evaluate outcomes is typically the sum of the players’ ordinal utilities.

The structure of an ordinal strategic game with equivalent preferences can be represented as follows:

  • Players: There are n players, denoted by i = 1, …, n.
  • Actions: Each player i has a set of actions, denoted by Ai.
  • Outcomes: An outcome of the game is a vector of actions, denoted by a = (a1, …, an), where ai ∈ Ai is the action chosen by player i.
  • Preferences: Each player i has a preference relation over the set of outcomes, denoted by Ri. The preference relation is assumed to be complete and transitive.
  • Utility: The utility of an outcome to player i is denoted by ui(a). The utility function is assumed to be ordinal, meaning that it only represents the relative ranking of outcomes and not their absolute differences.

The following table summarizes the key elements of an ordinal strategic game with equivalent preferences:

Element Description
Players The set of players in the game
Actions The set of actions available to each player
Outcomes The set of possible outcomes of the game
Preferences The preference relation of each player over the set of outcomes
Utility The utility of an outcome to each player

The payoff function for each player is a function that maps the set of actions to the player’s utility. The payoff function is typically derived from the player’s preference relation.

The social welfare function for an ordinal strategic game with equivalent preferences is a function that maps the set of actions to the sum of the players’ utilities. The social welfare function is used to evaluate the outcomes of the game and determine the optimal outcome.

Example:

Consider a game with two players, A and B, who have the following preferences:

Outcome Player A Player B
(a1, a1) 1 1
(a1, a2) 2 0
(a2, a1) 0 2
(a2, a2) 3 3

The payoff function for each player is:

Action Player A Player B
a1 1 1
a2 3 3

The social welfare function is:

Outcome Player A Player B Sum
(a1, a1) 1 1 2
(a1, a2) 2 0 2
(a2, a1) 0 2 2
(a2, a2) 3 3 6

The optimal outcome of the game is (a2, a2), which has the highest social welfare value of 6.

Question 1:
What are the key characteristics of ordinal strategic games with equivalent preferences?

Answer:
Ordinal strategic games with equivalent preferences are games where:
– Players’ preferences are ordinal, meaning they can only rank alternatives from best to worst.
– Players’ preferences are equivalent, meaning that they all have the same ordering of alternatives.
– Players’ strategies are pure, meaning they choose one action from a set of possible actions.
– Players’ payoffs are determined by the actions of all players, and the player with the highest payoff wins.

Question 2:
How do ordinal strategic games with equivalent preferences differ from cardinal strategic games?

Answer:
In ordinal strategic games with equivalent preferences, players’ preferences are ordinal, while in cardinal strategic games, players’ preferences are cardinal. This means that in cardinal strategic games, players can not only rank alternatives from best to worst, but they can also specify the magnitude of their preferences.

Question 3:
What are some applications of ordinal strategic games with equivalent preferences?

Answer:
Ordinal strategic games with equivalent preferences have been used to model a variety of real-world scenarios, including:
– Voting systems, where voters rank candidates from best to worst.
– Auctions, where bidders rank items from most to least valuable.
– Matching markets, where workers and firms rank each other from most to least desirable.

And that’s a wrap for today’s dive into ordinal strategic games with equivalent preferences! We hope you enjoyed this little mind-bender. If you’re still hungry for more game theory goodies, be sure to stick around. We’ve got plenty more where that came from. Thanks for reading, folks!

Leave a Comment