3$, there are precisely three $\preceq$-minimal mechanisms $M_{G}$ in $\mathfrak{M}(m)$, where $G$ corresponds to the star, cycle and complete graphs. The star mechanism has a distinguished commodity -- the money -- that serves as the sole medium of exchange and mediates trade between decentralized markets for the other commodities. Our main result is that, for any weights $\lambda,\mu>0,$ the star mechanism is the unique minimizer of $\lambda\tau(M)+\mu\pi(M)$ on $\mathfrak{M}(m)$ for large enough $m$."> 3$, there are precisely three $\preceq$-minimal mechanisms $M_{G}$ in $\mathfrak{M}(m)$, where $G$ corresponds to the star, cycle and complete graphs. The star mechanism has a distinguished commodity -- the money -- that serves as the sole medium of exchange and mediates trade between decentralized markets for the other commodities. Our main result is that, for any weights $\lambda,\mu>0,$ the star mechanism is the unique minimizer of $\lambda\tau(M)+\mu\pi(M)$ on $\mathfrak{M}(m)$ for large enough $m$.">
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Money as Minimal Complexity

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  • Pradeep Dubey
  • Siddhartha Sahi
  • Martin Shubik
Abstract
We consider mechanisms that provide traders the opportunity to exchange commodity $i$ for commodity $j$, for certain ordered pairs $ij$. Given any connected graph $G$ of opportunities, we show that there is a unique mechanism $M_{G}$ that satisfies some natural conditions of "fairness" and "convenience". Let $\mathfrak{M}(m)$ denote the class of mechanisms $M_{G}$ obtained by varying $G$ on the commodity set $\left\{1,\ldots,m\right\} $. We define the complexity of a mechanism $M$ in $\mathfrak{M(m)}$ to be a certain pair of integers $\tau(M),\pi(M)$ which represent the time required to exchange $i$ for $j$ and the information needed to determine the exchange ratio (each in the worst case scenario, across all $i\neq j$). This induces a quasiorder $\preceq$ on $\mathfrak{M}(m)$ by the rule \[ M\preceq M^{\prime}\text{if}\tau(M)\leq\tau(M^{\prime})\text{and}\pi(M)\leq\pi(M^{\prime}). \] We show that, for $m>3$, there are precisely three $\preceq$-minimal mechanisms $M_{G}$ in $\mathfrak{M}(m)$, where $G$ corresponds to the star, cycle and complete graphs. The star mechanism has a distinguished commodity -- the money -- that serves as the sole medium of exchange and mediates trade between decentralized markets for the other commodities. Our main result is that, for any weights $\lambda,\mu>0,$ the star mechanism is the unique minimizer of $\lambda\tau(M)+\mu\pi(M)$ on $\mathfrak{M}(m)$ for large enough $m$.

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  • Pradeep Dubey & Siddhartha Sahi & Martin Shubik, 2015. "Money as Minimal Complexity," Papers 1512.02317, arXiv.org, revised Dec 2015.
    • Handle: RePEc:arx:papers:1512.02317
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    JEL classification:

    • C70 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - General
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • C79 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Other
    • D44 - Microeconomics - - Market Structure, Pricing, and Design - - - Auctions
    • D63 - Microeconomics - - Welfare Economics - - - Equity, Justice, Inequality, and Other Normative Criteria and Measurement
    • D82 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Asymmetric and Private Information; Mechanism Design

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