# Abstract

We provide insights into (Bach, Christian W., and Andrés Perea’s. 2014. “Utility Proportional Beliefs.” *International Journal of Game Theory*: 1–22) concept of utility proportional beliefs and characterize its underlying reasoning process in normal-form games. Our analysis suggests that assumptions on the sensitivity to utilities influence players’ reasoning process and therefore the computations necessary to obtain a belief about the opponent. Under the assumption that more complex computations take longer to complete, we develop additional hypotheses about the players’ reaction times. These additional hypotheses allow for more rigorous testing of the concept than pure accordance with predictions. Using (Nauerz, Christian T., Marion Collewet, and Frauke Meyer. 2015. “Explaining Beliefs in Strictly Competitive One-shot Games.” *Working Paper*) data set we confirm our hypotheses about players’ reaction times, strengthening our trust in the concept.

# Acknowledgements:

This paper was previously known as “Understanding reasoning in games using utility proportional beliefs”.

I am grateful to Andrs Perea and Elias Tsakas, and to Christian Bach, Matthew Embrey, Marion Collewet and Frauke Meyer. I am also thankful for comments from two referees and the seminar participants at Maastricht University, the Erasmus University in Rotterdam and the conference participants at LOFT, ROAM and SYME 2013.

# Appendix

## Proof of Proposition 1.

First, remember that *k-fold belief in *. For the case of common belief in

*, we let*λ -utility-proportional-beliefs

for some .

## Lemma 3.

*It holds that**for all**for all*

## proof.

We prove Lemma 3 by induction on *k.* Algebraically, the composed function

Hence, we have for

For some

Now substitute

which is what we wanted to prove.

## Proof of Lemma 2.

According to Lemma 3 we have that

for every

Since BP showed in their Theorem 1 that the iterative application of , and in their Theorem 2 that these beliefs must be unique in the two player case,

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**Published Online:**2018-06-27

© 2018 Walter de Gruyter GmbH, Berlin/Boston