<p>I am currently studying game theory and have just begun looking at repeated strategic games.</p>
<p>In my lecture notes, it states that &quot;preferences are unique up to an affine transformation&quot;, which I understand. However, it then proceeds to talk about the implications of that statement for repeated games, namely,  &quot;different payoffs for the same strategic game may generate different preferences for the repeated game&quot;. Am I correct in surmising that this is to do with the fact that different payoffs imply a non-affine transformation and hence the preferences change?</p>

CC BY-SA 4.0
2021-02-27T13:37:26.407Z
The statement by itself is not sufficiently clear. Is there some additional context before and after?
Michael Greinecker
CC BY-SA 4.0
2021-02-27T13:41:52.083Z
Thank you for your reply. There isn't much context prior, but afterwards, the notes give an example of two versions of a Prisoners' dilemma (the same strategic game with different payoffs) whereby in the former when repeated the aggregate payoff is greater for colluding twice as opposed to alternating between confessing and not-confessing; whereas for the latter the aggregate payoff is lower for colluding twice compared to alternating between confessing and not-confessing because the payoffs for confessing have been increased
KurisuUwU
CC BY-SA 4.0
2021-02-27T13:46:36.457Z
I think what this might be about is that payoffs are defined on all strategy-profiles and randomization thereover. But each player can only affect what they are doing themselves.  There are payoff-changes that leave the individual decision problem unchanged. For example, if something is added to the payoff function that only depends on the other players action, the decision-problem does not change. This way, you can even rewrite the one-shot prisoners dilemma so that both players face the same decision-problem but both defecting is Pareto-better than both cooperating.
Michael Greinecker
CC BY-SA 4.0
2021-02-27T13:53:07.133Z
I see, so are you saying that since utility preferences in game theory are to do with a system of decisions that involve other players, given an affine transformation over which a player has no control will not change their preferences? i.e. are the preferences robust to affine transformations in repeated games just like with normal utility function problems ?
KurisuUwU
CC BY-SA 4.0
2021-02-27T13:58:29.657Z
It is not a positive affine transformation of the payoff-function, but it is a positive affine transformation for fixed behavior of the other player(s).
Michael Greinecker
CC BY-SA 4.0
2021-02-27T14:01:46.270Z
Ah I think I'm getting there, so given a positive affine transformation of the behaviour of other player(s) the behaviour of the first player will not change
KurisuUwU
CC BY-SA 4.0
2021-02-27T14:11:01.790Z
Not quite. Consider a two-player game with action spaces $A_i$ and payoff-functions $u_i:A_1\times A_2\to\mathbb{R}$ with $i=1,2$. Consider a payoff-function $v_1:A_1\times A_2\to\mathbb{R}$ of player $1$ of the form $v_1(a_1,a_2)=\alpha(a_2)u_1(a_1,a_2)+\beta(a_2)$ with $\alpha:A_2\to\mathbb{R}_{++}$ a function with strictly positive values and $\beta:A_2\to\mathbb{R}$ any function. Then, $v_1$ is not a positive affine transformation of $u_1$. but whatever behavior player $2$ has, the "induced" payoff functions on $A_1$ by $u_1$ and $v_1$ are positive affine transformations of each other.
Michael Greinecker
CC BY-SA 4.0
2021-02-27T14:22:00.123Z
Sorry, I'm with you until you mention that v1 is not a positive affine transformation of u1, could you elaborate a bit further on why that is so?
KurisuUwU
CC BY-SA 4.0
2021-02-27T14:24:56.760Z
What you do not have (well, not necessarily) is some $\alpha>0$ and some $\beta$ such that $v_1(a_1,a_2)=\alpha u_1(a_1,a_2)+\beta$ for all $a_1$ and (!) $a_2$.
Michael Greinecker
CC BY-SA 4.0
2021-02-27T14:27:12.337Z
The last question, thank you for helping me out on this. Is that because alpha is a function of a2, i.e. the specific action of player 2?
KurisuUwU
CC BY-SA 4.0
2021-02-27T14:28:35.057Z
Exactly, for a positive affine transformation $\alpha$ and $\beta$ have to be numbers, not functions.
Michael Greinecker
<microeconomics><game-theory><utility>
Von Neumann–Morgenstern implications for repeated strategic games
KurisuUwU
# Von Neumann–Morgenstern implications for repeated strategic games

(Asked by: username_0 on 2021-02-27)

I am currently studying game theory and have just begun looking at repeated strategic games.

In my lecture notes, it states that "preferences are unique up to an affine transformation", which I understand. However, it then proceeds to talk about the implications of that statement for repeated games, namely, "different payoffs for the same strategic game may generate different preferences for the repeated game". Am I correct in surmising that this is to do with the fact that different payoffs imply a non-affine transformation and hence the preferences change?

**Tags:** microeconomics, game-theory, utility
> **Comments:**
> **username_1**: The statement by itself is not sufficiently clear. Is there some additional context before and after?  
> **username_0**: Thank you for your reply. There isn't much context prior, but afterwards, the notes give an example of two versions of a Prisoners' dilemma (the same strategic game with different payoffs) whereby in the former when repeated the aggregate payoff is greater for colluding twice as opposed to alternating between confessing and not-confessing; whereas for the latter the aggregate payoff is lower for colluding twice compared to alternating between confessing and not-confessing because the payoffs for confessing have been increased  
> **username_1**: I think what this might be about is that payoffs are defined on all strategy-profiles and randomization thereover. But each player can only affect what they are doing themselves.  There are payoff-changes that leave the individual decision problem unchanged. For example, if something is added to the payoff function that only depends on the other players action, the decision-problem does not change. This way, you can even rewrite the one-shot prisoners dilemma so that both players face the same decision-problem but both defecting is Pareto-better than both cooperating.  
> **username_0**: I see, so are you saying that since utility preferences in game theory are to do with a system of decisions that involve other players, given an affine transformation over which a player has no control will not change their preferences? i.e. are the preferences robust to affine transformations in repeated games just like with normal utility function problems ?  
> **username_1**: It is not a positive affine transformation of the payoff-function, but it is a positive affine transformation for fixed behavior of the other player(s).  
> **username_0**: Ah I think I'm getting there, so given a positive affine transformation of the behaviour of other player(s) the behaviour of the first player will not change  
> **username_1**: Not quite. Consider a two-player game with action spaces $A_i$ and payoff-functions $u_i:A_1\times A_2\to\mathbb{R}$ with $i=1,2$. Consider a payoff-function $v_1:A_1\times A_2\to\mathbb{R}$ of player $1$ of the form $v_1(a_1,a_2)=\alpha(a_2)u_1(a_1,a_2)+\beta(a_2)$ with $\alpha:A_2\to\mathbb{R}_{++}$ a function with strictly positive values and $\beta:A_2\to\mathbb{R}$ any function. Then, $v_1$ is not a positive affine transformation of $u_1$. but whatever behavior player $2$ has, the "induced" payoff functions on $A_1$ by $u_1$ and $v_1$ are positive affine transformations of each other.  
> **username_0**: Sorry, I'm with you until you mention that v1 is not a positive affine transformation of u1, could you elaborate a bit further on why that is so?  
> **username_1**: What you do not have (well, not necessarily) is some $\alpha>0$ and some $\beta$ such that $v_1(a_1,a_2)=\alpha u_1(a_1,a_2)+\beta$ for all $a_1$ and (!) $a_2$.  
> **username_0**: The last question, thank you for helping me out on this. Is that because alpha is a function of a2, i.e. the specific action of player 2?  
> **username_1**: Exactly, for a positive affine transformation $\alpha$ and $\beta$ have to be numbers, not functions.  

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