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

Question

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.

Answer 1

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 $\mu_i(a) \le \mu_i(b)$.