# "Nash equilibrium" involving altruists, self-sacrificiers, lovers, haters and teams...

In a Nash equilibrium, no player has anything to gain by changing only their own strategy. And we know that Nash has proved that there is a Nash equilibrium for every finite game.

Players are usually assumed to be rational agents in game theory, i.e. they seek to maximize their own interests. Then may I asked the question that when "non-rational-agents" were involved in, is there necessarily a equilibrium in a similar way to Nash equilibrium still?

We call a player who seeks to maximize others' interests (but not her own) a "altruist"；

A player who seeks to maximize the collection's interest (not matter how much she can get for herself) is called a "collectivist";

A player $$i$$ who only seeks to maximize another player $$k$$'s interest (maybe because player $$i$$ loves player $$k$$) is called an "admirer";

By contrast, a player $$i$$ who seeks to minimalize another player $$k$$'s interest is called a "hater";

A player who seeks to minimalize her own interest is called a "self-sacrificier";

A set of players seeking to maximize their own group's interest are called a "team" (and in the special case a set of two players, they are called "lovers") .

Correspondingly,

when "altruists" were involved in, is there necessarily a equilibrium in a similar way to Nash equilibrium still?

When "collectivist" were involved in, is there necessarily a equilibrium in a similar way to Nash equilibrium still?

......

I apologize for my ignorance if there are any problems.

## 1 Answer

After learning a little bit game theory i think the answer is yes

the definition of a stragetic game is：

• a finite set $$N$$ (the set of players)
• for each player $$i\in N$$ a nonempty set $$A_i$$ (the set of actions available to player $$i$$)
• for each player $$i\in N$$ a preference relation $$\ge_i$$ on $$A=\times_{j\in N}A_j$$(the preference relation of player $$i$$)

because of the high level of abstraction of this model, it could be applied to a wide variety of situations，including the cases above.

E.g. In the case of self-sacrificier, we could define a payoff funtion $$\mu_i:A\rightarrow\mathbb{R}$$, and let the preference relation $$\ge_i$$ for a self-sacrificier $$i$$ be that $$a \ge_i b$$ whenever $$\mu_i(a) \le \mu_i(b)$$.