<p>It does appear to be a mistake in this pdf version of the book, where the authors are dealing with rewards: The regret should be the reward that would have been gained for playing s minus the actual gain.</p>

<p>Note that often this is set up with losses (rather than rewards) so that the regret would be the incurred loss minus the loss that would have been incurred under some pure strategy.</p>

<p>For a given pure strategy s, the regret for not having played s should be 0 if one's choice gave the same payoff as s, positive if it was worse (i.e., lower in a rewards setting, higher in a losses setting) than s, and negative if it was better than s.</p>

<p>Commonly one defines <strong>the</strong> regret as the regret for not having played the <strong>best</strong> pure strategy (given the choices of others). This will be non-negative, and 0 if and only if the current choice is optimal.</p>

CC BY-SA 3.0
2014-05-25T06:55:42.810Z
2014-05-25T14:20:22.810Z
2014-05-25T14:20:22.810Z
Rahul Savani
<p>I am reading this <a href="http://www.masfoundations.org/download.html" rel="nofollow">book</a>. In chapter 7, section 7.5 page 240 (in the pdf), the authors defined (definition 7.5.1) the <em>regret</em> as being the difference between the average per-period reward the agent received up until time $t$ and the average per-period reward the agent would have received up until time $t$ had he played pure strategy $s$ instead, i.e., $\mathrm{Regret}=\alpha^t-\alpha^t(s)$.</p>

<p>I do not get this definition of <em>regret</em>. For me it looks the other way around, i.e., $\mathrm{Regret}=\alpha^t(s)-\alpha^t$. I also looked for definitions in the internet and I am really lost now.</p>

<p>Can you please explain to me what I am missing here? Could the authors be wrong in the definition (since it is uncorrected manuscript)?</p>

<reference-request><machine-learning><gt.game-theory>
How to define the _regret_ in multiagent systems? Any good definition please?
Jika
# How to define the _regret_ in multiagent systems? Any good definition please?

(Asked by: username_0 on 2014-05-24)

I am reading this [book](http://www.masfoundations.org/download.html). In chapter 7, section 7.5 page 240 (in the pdf), the authors defined (definition 7.5.1) the *regret* as being the difference between the average per-period reward the agent received up until time $t$ and the average per-period reward the agent would have received up until time $t$ had he played pure strategy $s$ instead, i.e., $\mathrm{Regret}=\alpha^t-\alpha^t(s)$.

I do not get this definition of *regret*. For me it looks the other way around, i.e., $\mathrm{Regret}=\alpha^t(s)-\alpha^t$. I also looked for definitions in the internet and I am really lost now.

Can you please explain to me what I am missing here? Could the authors be wrong in the definition (since it is uncorrected manuscript)?

**Tags:** reference-request, machine-learning, gt.game-theory


---

## 1 answers

### Answer by username_1 on 2014-05-25. Score: 2 | Accepted answer

It does appear to be a mistake in this pdf version of the book, where the authors are dealing with rewards: The regret should be the reward that would have been gained for playing s minus the actual gain.

Note that often this is set up with losses (rather than rewards) so that the regret would be the incurred loss minus the loss that would have been incurred under some pure strategy.

For a given pure strategy s, the regret for not having played s should be 0 if one's choice gave the same payoff as s, positive if it was worse (i.e., lower in a rewards setting, higher in a losses setting) than s, and negative if it was better than s.

Commonly one defines **the** regret as the regret for not having played the **best** pure strategy (given the choices of others). This will be non-negative, and 0 if and only if the current choice is optimal.


---

