The Daily Thoth proposes that Xen is in fact not a Zero-Sum Game.
- ?? Close your eyes for a moment and imagine a world not bound by traditional choices.
- What if, in our example, instead of choosing between Movie ??? and Movie ???, both players were deciding whether or not to mint Xen?
- What impact might that decision have on their strategy and the overarching equilibrium?
- Would the stability of their choices be affected, or would they still gravitate towards a consistent, optimal strategy?
Ponder this and let the implications sink in……
The Nash Equilibrium is a fundamental concept in game theory that describes a state in which all players in a game choose a strategy that is the best response to the strategies chosen by all other players. This means that no player can improve their outcome by changing their strategy alone, assuming all other players' strategies remain the same.
The Nash Equilibrium is widely used in economics, political science, and other fields to analyze a wide range of scenarios, from pricing strategies in markets to international conflicts between nations. One famous example is the prisoner's dilemma, where two suspects must decide whether to confess or remain silent, with the Nash Equilibrium being both suspects confessing.
Imagine you and a friend are trying to decide which movie to watch tonight. You each have a preference for one of two movies: Movie ??? or Movie ???. You both have the option to either speak up and express your preference or keep quiet.
Scenario:
If we assume that both you and your friend are rational and want to maximize their payoff, then we can determine the Nash Equilibrium by looking at each player's best response to the other's strategy.
You have a beautiful mind…..
Example: If you assume your friend will choose ?? Movie - ???, then your best response is to also choose ?? Movie - ??? because it gives you a higher payoff of 2. If you assume your friend will choose Movie B, then your best response is to choose Movie A because it still gives you a higher payoff of 1 than Movie B, which only gives you a payoff of 0.
If your friend assumes you will choose Movie ???, then their best response is also to choose Movie ??? because it gives them a higher payoff of 1??. If your friend assumes you will choose Movie ???, then their best response is to choose Movie ??? because it gives them a higher payoff of ??.
So, the Nash Equilibrium occurs when both you and your friend choose Movie ???. In this case, neither of you can improve your payoff by changing your strategy, given the other player's strategy. The Nash Equilibrium is a stable state in which both players are doing the best they can, given the other player's choice.
u1(s1*, s2*) >= u1(s1, s2*) for all s1 in S1 and u2(s1*, s2*) >= u2(s1*, s2) for all s2 in S2
Let S1 be the set of strategies available to player 1?? and S2 be the set of strategies available to player 2?? Let u1(s1, s2) be the payoff or utility function for player 1?? when they play strategy s1 and player 2?? plays strategy s2. Similarly, let u2(s1, s2) be the payoff function for player 2?? when they play strategy s2 and player 1?? plays strategy s1.
Then, the Nash Equilibrium is a pair of strategies (s1*, s2*) such that:
u1(s1*, s2*) >= u1(s1, s2*) for all s1 in S1 and u2(s1*, s2*) >= u2(s1*, s2) for all s2 in S2
Player 1?? is playing their best response given player 2??'s strategy, and player 2?? is playing their best response given player 1??'s strategy. No player can improve their payoff by unilaterally changing their strategy, assuming the other player's strategy remains the same.
Mint $Xen
In the example I gave earlier, the Nash Equilibrium occurs when both players choose Movie ??? because it satisfies the conditions of the above equation. If one player deviates from this strategy, their payoff will decrease, so it's a stable state in which neither player has an incentive to change their strategy.
