<p>Yes, a whole book has been written on <a href="https://press.princeton.edu/titles/7517.html" rel="nofollow noreferrer">Behavioral Game Theory</a>. </p>

<p>More specifically, standard solution concept such as Nash equilibrium requires that players <strong><em>best respond</em></strong> to a <strong><em>correct belief</em></strong> about other players' moves. The following are examples that relax one of these cognitive restrictions:</p>

<ul>
<li><a href="https://en.wikipedia.org/wiki/Quantal_response_equilibrium" rel="nofollow noreferrer">Quantal response equilibrium</a> allows for the possibility that players do not always choose a best response. Instead, they "quantal respond" (or best respond with mistakes) to a correct belief about others' quantal response. </li>
<li><a href="https://en.wikipedia.org/wiki/Cognitive_hierarchy_theory" rel="nofollow noreferrer">Level-<em>k</em> or Cognitive Hierarchy Theory</a> relaxes the correctness of beliefs assumption, and allows players to best respond to a set of mutually inconsistent beliefs. </li>
<li><a href="https://ac.els-cdn.com/0899825689900055/1-s2.0-0899825689900055-main.pdf?_tid=5119d6d2-d49c-11e7-92e6-00000aab0f6c&amp;acdnat=1511915530_bbe87353c0e762ef52a7b113f865977f" rel="nofollow noreferrer">Psychological games</a> allow players to derive utility directly from beliefs (and belief-dependent emotions). Relevant solution concept include <a href="http://www.u.arizona.edu/~martind1/Papers-Documents/atosr.pdf" rel="nofollow noreferrer">sequential reciprocity equilibrium</a> and <a href="https://people.hss.caltech.edu/~camerer/NYU/07-Rabin.pdf" rel="nofollow noreferrer">Rabin's fairness equilibrium</a>. </li>
</ul>

<p>For a complete review of models of behavioral game theory, see Chapter 13 of <a href="https://rads.stackoverflow.com/amzn/click/B01N7D2D20" rel="nofollow noreferrer">Dhami (2016)</a>.</p>

<p>For a specific example of a game with mixed players, see this paper on <a href="https://www.researchgate.net/publication/266318534_Robust_protection_of_fisheries_with_COmPASS" rel="nofollow noreferrer">robust protection of fisheries</a>. A more comprehensive treatment of mixed games is available in Rani Spiegler's <a href="http://www.tau.ac.il/~rani/" rel="nofollow noreferrer">Bounded Rationality and Industrial Organization</a> per Michael Greinecker's comment on the original question.</p>

CC BY-SA 3.0
2017-11-29T00:57:04.717Z
After checking out those links, I'm still a little unclear if those behavioral game frameworks allow some players to be standard rational players and other players to be behavioral players (in the same game). For instance, in the Cognitive Hierarchy Theory wiki page it sounds like all the players have finite depth reasoning, so there would be no truly rational player in that game.
ajkeith
CC BY-SA 3.0
2017-11-29T01:03:23.530Z
I edited the original post to better reflect that the question is about mixing rational and irrational players
ajkeith
CC BY-SA 3.0
2017-11-29T02:44:30.163Z
@ajkeith: Perhaps my [answer to this question](https://economics.stackexchange.com/a/18854/42) may be an example?
Herr K.
CC BY-SA 3.0
2017-11-29T08:08:21.353Z
@ajkeith If everyone plays a best response, everyone is as rational as can be and can reason as deeply as possible. Everyone being rational does not imply that everyone believes everyone to be rational.
Michael Greinecker
CC BY-SA 3.0
2017-11-29T14:56:11.177Z
@HerrK.: Yes, that's a great example - I was hoping for some references that are focused on those types of games with mixed players. Michael Greinecker's comment on the original question is the exact type of answer I was looking for.
ajkeith
CC BY-SA 3.0
2017-11-29T00:41:52.603Z
2017-11-29T16:07:17.283Z
2017-11-29T16:07:17.283Z
Herr K.
<p>Are there game theoretic models where one player is rational and the other is irrational (i.e., plays with behavioral limitations)?</p>

<p>The motivation for this question is that behavioral economics has documented how people often make choices that don't maximize expected utility, but those irrational people might be interacting with near-rational computers or organizations. I've never seen a game theory model that has both an expected utility-maximizing player and a behavioral player. </p>

CC BY-SA 3.0
2017-11-29T00:09:34.047Z
There is a literature on expected-profit maximizing firms dealing with behavioral consumers.Take a look at Ran Spiegler's excellent book.
Michael Greinecker
<game-theory><behavioral-economics><bayesian-game><repeated-games>
Game theory with rational and irrational players
ajkeith
# Game theory with rational and irrational players

(Asked by: username_0 on 2017-11-28)

Are there game theoretic models where one player is rational and the other is irrational (i.e., plays with behavioral limitations)?

The motivation for this question is that behavioral economics has documented how people often make choices that don't maximize expected utility, but those irrational people might be interacting with near-rational computers or organizations. I've never seen a game theory model that has both an expected utility-maximizing player and a behavioral player.

**Tags:** game-theory, behavioral-economics, bayesian-game, repeated-games
> **Comments:**
> **username_1**: There is a literature on expected-profit maximizing firms dealing with behavioral consumers.Take a look at Ran Spiegler's excellent book.  

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## 1 answers

### Answer by username_2 on 2017-11-29. Score: 4 | Accepted answer

Yes, a whole book has been written on [Behavioral Game Theory](https://press.princeton.edu/titles/7517.html).

More specifically, standard solution concept such as Nash equilibrium requires that players ***best respond*** to a ***correct belief*** about other players' moves. The following are examples that relax one of these cognitive restrictions:

* [Quantal response equilibrium](https://en.wikipedia.org/wiki/Quantal_response_equilibrium) allows for the possibility that players do not always choose a best response. Instead, they "quantal respond" (or best respond with mistakes) to a correct belief about others' quantal response.
* [Level-*k* or Cognitive Hierarchy Theory](https://en.wikipedia.org/wiki/Cognitive_hierarchy_theory) relaxes the correctness of beliefs assumption, and allows players to best respond to a set of mutually inconsistent beliefs.
* [Psychological games](https://ac.els-cdn.com/0899825689900055/1-s2.0-0899825689900055-main.pdf?_tid=5119d6d2-d49c-11e7-92e6-00000aab0f6c&acdnat=1511915530_bbe87353c0e762ef52a7b113f865977f) allow players to derive utility directly from beliefs (and belief-dependent emotions). Relevant solution concept include [sequential reciprocity equilibrium](http://www.u.arizona.edu/~martind1/Papers-Documents/atosr.pdf) and [Rabin's fairness equilibrium](https://people.hss.caltech.edu/~camerer/NYU/07-Rabin.pdf).

For a complete review of models of behavioral game theory, see Chapter 13 of [Dhami (2016)](https://rads.stackoverflow.com/amzn/click/B01N7D2D20).

For a specific example of a game with mixed players, see this paper on [robust protection of fisheries](https://www.researchgate.net/publication/266318534_Robust_protection_of_fisheries_with_COmPASS). A more comprehensive treatment of mixed games is available in Rani Spiegler's [Bounded Rationality and Industrial Organization](http://www.tau.ac.il/~rani/) per Michael Greinecker's comment on the original question.

> **Comments:**
> **username_0**: After checking out those links, I'm still a little unclear if those behavioral game frameworks allow some players to be standard rational players and other players to be behavioral players (in the same game). For instance, in the Cognitive Hierarchy Theory wiki page it sounds like all the players have finite depth reasoning, so there would be no truly rational player in that game.  
> **username_0**: I edited the original post to better reflect that the question is about mixing rational and irrational players  
> **username_2**: @ajkeith: Perhaps my [answer to this question](https://economics.stackexchange.com/a/18854/42) may be an example?  
> **username_1**: @ajkeith If everyone plays a best response, everyone is as rational as can be and can reason as deeply as possible. Everyone being rational does not imply that everyone believes everyone to be rational.  
> **username_0**: @HerrK.: Yes, that's a great example - I was hoping for some references that are focused on those types of games with mixed players. Michael Greinecker's comment on the original question is the exact type of answer I was looking for.  

---

