# American Institute of Mathematical Sciences

doi: 10.3934/jdg.2022007
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## A Bayesian equilibrium for simultaneous first-price auctions for complementary goods and quasi-linear bids

 1 Universidad Autónoma de San Luis Potosí, School of Science, Av. Chapultepec No. 1570, Mexico 2 Universidad Autónoma de San Luis Potosí, School of Economics, Av. Pintores S/N, Mexico

* Corresponding author: karla.zarur@uaslp.mx

Received  August 2021 Revised  February 2022 Early access March 2022

Fund Project: The first author is supported by CONACyT grant 2019-000002-01NACF-06046

This article shows a Symmetrical Bayesian Nash Equilibrium in a context of $m$ simultaneous first-price sealed-bid auctions and $n$ bidders for complementary goods. We consider that the individual valuations of the $m$ goods are common knowledge and identical among bidders and if the whole set of goods is gained, a private independent extra profit is obtained by the winner. For relaxing and solving the so-many mathematical complications involved in the general case we followed a classical methodology and proposed a particular bidding function that implies linear separability. Under this assumptions we obtain a Symmetric Bayesian Nash Equilibrium which functional form implies the classic quasi-linear property for bivariate functions. On addition, we compare the seller expected revenue between auctioning the complete set in one single first-price sealed-bid auction versus auctioning each item in $m$ simultaneous first-price sealed-bid auctions.

Citation: Karla Flores-Zarur, William Olvera-Lopez. A Bayesian equilibrium for simultaneous first-price auctions for complementary goods and quasi-linear bids. Journal of Dynamics and Games, doi: 10.3934/jdg.2022007
##### References:
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show all references

##### References:
 [1] J. C. Harsanyi, Games with incomplete information played by bayesian players, Ⅰ–Ⅲ part Ⅰ. The basic model, Management Science, 14 (1967), 159-182.  doi: 10.1287/mnsc.14.3.159. [2] O. Morgenstern, A. Tucker and W. Vickrey, Auctions and bidding games, in Recent Advances in Game Theory, Princeton University Press, Princeton, 1962. [3] P. R. Milgrom and R. J. Weber, A theory of auctions and competitive bidding, Econometrica, 50 (1982), 1089-1122.  doi: 10.2307/1911865. [4] H. P. Young and Z. Shmuel, Handbook of game theory with economic applications, Elsevier, 4 (2015). [5] P. L. Lorentziadis, Optimal bidding in auctions from a game theory perspective, European Journal of Operational Research, 248 (2016), 347-371.  doi: 10.1016/j.ejor.2015.08.012. [6] V. Krishna and R. W. Rosental, Simultaneous auctions with synergies, Games and Economic Behavior, 17 (1996), 1-31.  doi: 10.1006/game.1996.0092. [7] R. W. Rosental and R. Wang, Simultaneous auctions with synergies and common values, Games and Economic Behavior, 17 (1996), 32-55.  doi: 10.1006/game.1996.0093. [8] B. Szentes and R. W. Rosenthal, Three-object two-bidder simultaneous auctions: Chopsticks and tetrahedra, Games and Economic Behavior, 44 (2003), 114-133.  doi: 10.1016/S0899-8256(02)00530-4. [9] B. Szentes, Two-object two-bidder simultaneous auctions, International Game Theory Review, 9 (2007), 483-493.  doi: 10.1142/S0219198907001552. [10] H. Etzion, E. Pinker and A. Seidmann, Analyzing the simultaneous use of auctions and posted prices for online selling, Manufacturing and Service Operations Management, 8 (2006), 68-91.  doi: 10.1287/msom.1060.0101. [11] J. C. Harsanyi, Games with incomplete information played by bayesian players, Ⅰ–Ⅲ part Ⅱ. Bayesian equilibrium points, Management Science, 14 (1968), 320-334.  doi: 10.1287/mnsc.14.5.320.
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