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Beyond Coincidence: The Reasoning Process Underlying Utility Proportional Beliefs Process

Christian Tobias Nauerz

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.

JEL Classification: C72

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 Pik denotes the set of beliefs generated for player i in round k of the algorithm. Define for any two sets A,BPi0 and for all α[0,1] the set αA+(1α)B:=αa+(1α)b:aAand bB. Let pi(,λj) denote the function pi induced by the proportionality factor λj. BP’s Lemma 4 states that for the kth iteration of the algorithm there exists some belief piPi0 such that the convex combination αkPi0+(1αk){pi} contains all beliefs that a player can hold under up to k-fold belief in λ-utility-proportional-beliefs. For the case of common belief in λ-utility-proportional-beliefs, we let k go to infinity and obtain

(10)limkαkPi0+(1αk){pik}={pi}

for some piPi0 where pik denotes a belief pikPi0 such that PikαkPi0+(1αk){pik} holds. We observe that the convex combination converges to a singleton and hence to a unique fixed point. Now suppose piPi is the belief of i and pjPj is the belief of j for which eq. (10) holds. Recall, that by the construction of the algorithm it holds that Pik=pi(Pjk1,λj). Then it holds that pi(pj)=pi and pj(pi)=pj and hence it also holds that pi(pj(pi))=pi. Consequently, the mapping pipj has a unique fixed point in two-player games where the players’ are holding common belief in λ-utility-proportional-beliefs.

Lemma 3.

It holds that(pipj)k(pi)=(im+Gjin)+Sj(im+Gjin)++Sjk1(im+Gjin)+Sjkpifor allpiPi0for allk={1,,}.

proof.

We prove Lemma 3 by induction on k. Algebraically, the composed function pipj corresponds to

pi(pj(pi))=im+Gj(in+Gipi)=(im+Gjin)+Sjpi.

Hence, we have for k=1 that (pipj)(pi)=(im+Gjin)+Sjpi.

For some k>1, suppose the k1 previous steps were iteratively constructed by substituting pipj for pi at every step. Then we have

(pipj)k1(pi)=(im+Gjin)+Sj(im+Gjin)+Sj2(im+Gjin)++Sjk1pi.

Now substitute pi by (pipj)(pi) to obtain step k

(pipj)k(pi)=(im+Gjin)+Sj(im+Gjin)++Sjk1(im+Gjin)+Sjpi=(im+Gjin)+Sj(im+Gjin)++Sjk1(im+Gjin)+Sjkpi,

which is what we wanted to prove.

Proof of Lemma 2.

According to Lemma 3 we have that

(pipj)k(pi)=(im+Gjin)+Sj(im+Gjin)++Sjk1(im+Gjin)+Sjkpi,

for every kN and some piPi0. Factoring out (im+Gjin) yields

(pipj)k(pi)=(Im+Sj++Sjk1)(im+Gjin)+Sjkpi.

Since BP showed in their Theorem 1 that the iterative application of pi and pj yields exactly those beliefs that the players can hold under common belief in λ-utility-proportional-beliefs, and in their Theorem 2 that these beliefs must be unique in the two player case, (pipj)k(pi) must also converge to the unique belief for k. We then have limk(pipj)k(pi)=(Im+n=1Sjn)(im+Gjin), which shows that the infinite sum (Im+n=1Sjn) exists. It also follows that Sjk converges to the zero matrix for k since (pipj)k(pi) converges to the unique belief for k and since Sjk is the only element affected by k. As the infinite sum (Im+n=1Sjn) exists and Sjk converges to the zero matrix, it follows that the infinite sum (Im+n=1Sjn) always exists and that it converges to (Im+Sj)1.

References

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Published Online: 2018-06-27

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