<p>What you're asking seems to me just a matter of interpretation. Note that $\mathbb Z$ and $\mathbb Z_+$ have the same cardinality. So it makes no substantive difference which index set you use. </p>

<p>Moreover, in a continuation game with finite history, any strategy can be interpreted as a (possibly non-stationary) <a href="http://www.sciencedirect.com/science/article/pii/S0022053100927856" rel="nofollow"><strong>Markov strategy</strong></a> in a continuation game with infinitely long history. </p>

<p>If a history of infinite length is what you really care about, then may be looking into the literature of <a href="http://www.jstor.org/stable/2526948?seq=1#page_scan_tab_contents" rel="nofollow"><strong>continuous-time repeated game</strong></a> will provide some insight.</p>

CC BY-SA 3.0
2015-12-24T19:53:56.317Z
2015-12-24T19:53:56.317Z
Herr K.
<p>I do not know of any theory of repeated games without a beginning. However, some formalization may help regarding your question about stationary equilibria. I will answer your question with respect to the simple Nash equilibrium, but my remarks should also apply to more refined equilibrium concepts. Note first that we do not need to change our definition of a Nash equilibrium, as it is abstractly defined for any strategy space. </p>

<p>A Nash equilibrium in a repeated game with a starting point does not need to be stationary. For example, an infinitely repeated prisoners dilemma can have any sequence of "cooperate" and "defect" for both players given players do not discount future payoffs too much. This is the same for a game without a starting point. Suppose that the players have access to a binary randomization device to coordinate their strategies. Suppose they both play "cooperate" if the randomization device is 1 and "defect" otherwise. Any player deviating from this, will be punished with the other player playing "defect" for the remainder of the game. Now set the randomization device to probability $\frac{t^2}{t^2-1}$ of being 1 in period t. Clearly, the equilibrium strategies are non-stationary. However, players have no incentive to deviate as long as their time discounting is not too high.</p>

CC BY-SA 3.0
2015-09-25T06:56:30.050Z
2015-09-25T06:56:30.050Z
HRSE
<p>I'm interested to know results on repeated games indexing each stage game by $\mathbb Z$ which contrast with those indexed by $\mathbb Z_+$.</p>

<p>It seems to me this could be quite different from repeated games with a starting stage, because there's no need to concern ourselves about whether or not a node could be obtained. Is it the case that we only need to consider stationary equilibrium?</p>

CC BY-SA 3.0
2015-04-13T16:36:13.523Z
I don't really understand your idea. Do you mean the game starts (it has to start somehow, right?) with a random history? Do players start at the same time? If you have an equilibrium concept that has no different requirements for strategies on and off the equilibrium path the strategies should be the same in the game with or "without" a beginning.
The Almighty Bob
CC BY-SA 3.0
2015-04-13T16:40:38.320Z
Thank you for your comment. No, I mean, each player faces a history of infinite length. I can't give you a reference right now, but I remember somewhere, some author  mentioned, but didn't use,  this setting to justify the convergence to a stationary equilbrium, which was he mainly interested in.
Metta World Peace
CC BY-SA 3.0
2015-04-13T16:44:59.203Z
Maybe I see your point. But I'm not sure about how to define strategies on and off the equilibrium path in the first place, because it seems to me, we usually start to pinpoint them from the first stage.
Metta World Peace
CC BY-SA 3.0
2015-04-13T17:59:33.377Z
Not really an answer, but the literature on 'fictitious play' seems somehow relevant. See, e.g., here: http://web.stanford.edu/~jdlevin/Econ%20286/Learning.pdf
Ubiquitous
CC BY-SA 3.0
2015-04-13T18:01:03.040Z
@Ubiquitous, Thank you! I'll take a look.
Metta World Peace
<game-theory><reference-request><dynamic-games>
Repeated games without a beginning and an end
Metta World Peace
# Repeated games without a beginning and an end

(Asked by: username_0 on 2015-04-13)

I'm interested to know results on repeated games indexing each stage game by $\mathbb Z$ which contrast with those indexed by $\mathbb Z\_+$.

It seems to me this could be quite different from repeated games with a starting stage, because there's no need to concern ourselves about whether or not a node could be obtained. Is it the case that we only need to consider stationary equilibrium?

**Tags:** game-theory, reference-request, dynamic-games

> **Comments:**
> **username_1**: I don't really understand your idea. Do you mean the game starts (it has to start somehow, right?) with a random history? Do players start at the same time? If you have an equilibrium concept that has no different requirements for strategies on and off the equilibrium path the strategies should be the same in the game with or "without" a beginning.  
> **username_0**: Thank you for your comment. No, I mean, each player faces a history of infinite length. I can't give you a reference right now, but I remember somewhere, some author  mentioned, but didn't use,  this setting to justify the convergence to a stationary equilbrium, which was he mainly interested in.  
> **username_0**: Maybe I see your point. But I'm not sure about how to define strategies on and off the equilibrium path in the first place, because it seems to me, we usually start to pinpoint them from the first stage.  
> **username_2**: Not really an answer, but the literature on 'fictitious play' seems somehow relevant. See, e.g., here: http://web.stanford.edu/~jdlevin/Econ%20286/Learning.pdf  
> **username_0**: @username_2, Thank you! I'll take a look.  

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## 2 answers

### Answer by username_3 on 2015-12-24. Score: 2

What you're asking seems to me just a matter of interpretation. Note that $\mathbb Z$ and $\mathbb Z\_+$ have the same cardinality. So it makes no substantive difference which index set you use.

Moreover, in a continuation game with finite history, any strategy can be interpreted as a (possibly non-stationary) [**Markov strategy**](http://www.sciencedirect.com/science/article/pii/S0022053100927856) in a continuation game with infinitely long history.

If a history of infinite length is what you really care about, then may be looking into the literature of [**continuous-time repeated game**](http://www.jstor.org/stable/2526948?seq=1#page_scan_tab_contents) will provide some insight.


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### Answer by username_4 on 2015-09-25. Score: 0

I do not know of any theory of repeated games without a beginning. However, some formalization may help regarding your question about stationary equilibria. I will answer your question with respect to the simple Nash equilibrium, but my remarks should also apply to more refined equilibrium concepts. Note first that we do not need to change our definition of a Nash equilibrium, as it is abstractly defined for any strategy space.

A Nash equilibrium in a repeated game with a starting point does not need to be stationary. For example, an infinitely repeated prisoners dilemma can have any sequence of "cooperate" and "defect" for both players given players do not discount future payoffs too much. This is the same for a game without a starting point. Suppose that the players have access to a binary randomization device to coordinate their strategies. Suppose they both play "cooperate" if the randomization device is 1 and "defect" otherwise. Any player deviating from this, will be punished with the other player playing "defect" for the remainder of the game. Now set the randomization device to probability $\frac{t^2}{t^2-1}$ of being 1 in period t. Clearly, the equilibrium strategies are non-stationary. However, players have no incentive to deviate as long as their time discounting is not too high.


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