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In game theory, the aim of every player is to pick strategies that maximizes their payoff. This is done irrespective of the payoff of others. But in real life competitions, there arises a secondary aim to minimize the payoff of other players.

Consider this example of 2 students taking a test. Each student has two strategies, "Cheat"($C$) or "Be Honest"($H$). When both students cheat, the invigilator catches them and both get a payoff of -1 each. When both students take the test honestly, they both get a payoff of 1 each. But when one student cheats and the other takes the test honestly, the student who cheats gets away with a payoff of 2, while the honest student gets only a payoff of 1.

The Nash Equilibrium for this game is $(C,H)$ and $(H,C)$ because, neither student will move from this strategy as each ones individual payoff will decrease. But when the game is played iteratively, each of the equilibrium is preferred by each of the students, and both of them will aim to shift the game towards the equilibrium of their choice resulting in each of them selecting a different strategy to move towards their preferred equilibrium.

Consider the equilibrium $(C,H)$ which gives a payoff of 2 to player 1 (who cheats) and a payoff of 1 to player 2 (who takes the test honestly). Now, player 2's payoff will decrease if he/she chooses to switch strategy from $H$ to $C$ because $(C,C)$ will give a payoff of -1 to both players. However player 2 is jealous of player 1, because player 1 gets a higher payoff only because of player 2 not switching strategy from $H$ to $C$. In other words, player 1 takes advantage of player 2's honesty. Now, this gives motive to player 2 to switch his/her strategy from $H$ to $C$, resulting in the outcome of $(C,C)$ where both players get payoff value -1. Player 2 now hopes that player 1 will change strategy to get a higher payoff of 1 by taking the test honestly. This will result in the outcome $(H,C)$ which gives player 2 a higher payoff of 2 when compared to player 1's payoff of 1.

In the above example, we observe that the secondary aim of minimizing the payoff of the other player tends to disturb the Nash equilibrium of the original game.

What is the new equilibrium of this game?

Has this variation been considered in the study of game theory? Is there any existing study regarding the same?

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Some comments on this.

1) Your convergence interpretation changes the whole game. The game you describe is one shot - they play it once. If they were to play it several times in a row, then the game and consequently equilibria change. In most cases, more equilibria can be sustained in repeated games. This is basically what the Folk theorem says. For example, if you tell the other player: if you play $C$, then I will play $C$ forever whatever you do! (This is known as a grim trigger strategy). Then the other guy might never play $C$, because he is afraid of the "eternal" retribution. Thus, in repeated games you might be able to sustain honesty even with standard game theoretic tools.

2) Your description seems to resemble a war of attrition, which is a repeated game. If you are at $(C,C)$, it depends on "who can hold out longer", and the other guy will switch to being honest. The game is a bit different, because it stops as soon as someone plays $H$.

3) As you mention the fairness and payoffs of others. There are in fact equilibrium concepts that incorporate these things. The closest would be Rabin's 1993 "Fairness equilibrium". Briefly, if someone takes advantage of you, because he plays $C$ in the $(H,C)$ situation, then you might best respond by playing $C$ too. Although this reduces your payoff, you are compensated because "you get back at him for exploiting you". Very popular at the moment are social preferences like Fehr/Schmidt's 1999 inequity aversion or Bolton/Ockenfels preferences. They capture the idea that you care about what happens to the other guy when you do something, which is not captured in standard game theory with self-centered preferences.

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