Robust double auction mechanisms*  *  This work was supported by Basic Science
Research Program through the National Research Foundation of Korea (NRF)
funded by the Ministry of Education (2021R1$|$1A4A01059254) and by a Korea
University Grant (k2209871).

Kiho Yoon

Department of Economics, Korea University

145 Anam-ro, Seongbuk-gu, Seoul, Korea 02841

<EMAIL_ADDRESS>
http://econ.korea.ac.kr/~kiho

Abstract

We study the robust double auction mechanisms, that is, the double auction
mechanisms that satisfy dominant strategy incentive compatibility, ex-post
individual rationality and ex-post budget balance. We first establish that the
price in any robust mechanism does not depend on the valuations of the trading
players. We next establish that, with a non-bossiness assumption, the price in
any robust mechanism does not depend on players’ valuations at all, whether
trading or non-trading. Our main result is the characterization result that,
with a non-bossy assumption along with other assumptions on the properties of
the mechanism, the generalized posted mechanism in which a constant price is
posted for each possible set of traders is the only robust double auction
mechanism. We also show that, even without the non-bossiness assumption, it is
quite difficult to find a reasonable robust double auction mechanism other
than the generalized posted price mechanism.

Keywords: double auction, posted price, robust mechanism design, dominant
strategy, budget balance

JEL Classification: C72; D47; D82

1 Introduction

Double auctions are widely used in many two-sided market situations such as
stock exchanges as well as commodity markets. In a typical double auction, (i)
there are many sellers and many buyers who have private information about
their true valuations for the good, (ii) they simultaneously submit their
respective offers and bids to the mechanism (or the center, the mediator, the
clearing house, the social planner, etc.), and (iii) the mechanism determines
the volume and the terms of trade according to a pre-determined rule. Double
auctions approximate many other trading settings in the real world. They also
have great theoretical value in the study of price formation under private
information.

In this paper, we study the double auction mechanisms that satisfy dominant
strategy incentive compatibility, ex-post individual rationality and ex-post
budget balance. We call this family of double auction mechanisms as robust
double auction mechanisms. A special instance of the robust double auction
mechanisms is the robust bilateral trading mechanism for the environments in
which there is one seller who initially owns one indivisible good and there is
one buyer who is interested in obtaining the good. Hagerty and Rogerson (1987)
and Čopič and Ponsatí(2016) have shown that the only robust bilateral trading
mechanism is the posted price mechanism, i.e., the mechanism in which a price
is posted in advance and the seller and the buyer either trade at this price
or do not trade at all. The purpose of this paper is to obtain a similar
characterization for the general robust double auction mechanisms.

In the next section, we first establish that the price in any robust mechanism
does not depend on the valuations of the trading players. This result in
particular implies that a robust bilateral trading mechanism is a posted price
mechanism. In contrast, we show that a robust double auction mechanism may not
be a posted price mechanism when there are more than two players.1 1 Hence,
Hagerty and Rogerson’s conjecture (in page 97 of their paper) that the posted
price mechanism may be the only robust mechanism in ‘$n$-person trading
problems’ is incorrect. We next establish that, with a non-bossiness
assumption, the price in any robust mechanism does not depend on players’
valuations at all, whether trading or non-trading. Our main result is the
characterization result for the general double auction environments that, with
a non-bossy assumption along with other assumptions on the properties of the
mechanism, the generalized posted mechanism in which a constant price is
posted for each possible set of traders is the only robust double auction
mechanism. Section 3 further characterizes robust double auction mechanisms.
We show that, even without the non-bossiness assumption, it is quite difficult
to find a value-respecting robust double auction mechanism other than the
generalized posted price mechanism.2 2 Value-respecting mechanisms are those
mechanisms that select the traders according to the valuations. See Section 3
for details. Section 4 concludes.

There is a large literature on double auction mechanisms: Representative works
include Wilson (1985), Rustichini et al. (1994), Satterthwaite and Williams
(2002), Cripps and Swinkels (2006) and Satterthwaite et al. (2014).3 3 See
also Kojima and Yamashita (2017) and the references therein for double
auctions with interdependent values. Most of the papers in this literature
examine double auction mechanisms under the Bayesian incentive compatibility
postulate. In contrast, the current paper focuses on robust double auction
mechanisms. Robustness is desirable since the equilibrium behavior does not
depend on the fine details of the environments, such as players’ beliefs about
each other. We note that the only other double auction mechanisms that satisfy
dominant strategy incentive compatibility is McAfee’s (1992) mechanism and
Yoon’s (2001) modified Vickrey double auction mechanism, but these mechanisms
do not satisfy ex-post budget balance.

The papers similar in spirit but examining settings quite different from ours
are Barberà and Jackson (1995) and Miyagawa (2001). The former studies
classical exchange economies with many goods and shows that the only robust
mechanism, under some additional conditions, is the fixed-proportion anonymous
trading mechanism. The latter studies housing markets (i.e., trading
situations in which each player initially owns one object to trade) with money
and shows that the only robust mechanism, under some additional conditions, is
the fixed-price core mechanism.

The research on robust mechanism design is not limited to the trading
settings. Recently, Drexl and Kleiner (2015) and Shao and Zhou (2016) study
the robust allocation problem of assigning one indivisible private object to
one of two players, whereas Kuzmics and Steg (2017) study the robust public
good provision problem. On a more general level, there is a voluminous
literature concerning the common knowledge assumption and robust mechanism
design after the pioneering work of Bergemann and Morris (2005).4 4 See
Bergemann and Morris (2013) for an excellent introduction.

2 The main characterization

There is a set ${\cal S}=\\{1,\ldots,m\\}$ of sellers and a set ${\cal
B}=\\{1,\ldots,n\\}$ of buyers for a good. Each seller owns one indivisible
unit of the good to sell, and each buyer wants to buy at most one unit of the
good. Hence, there are $m$ units of the good available. Seller $i$’s privately
known valuation for the good is denoted by $s_{i}$, and buyer $j$’s privately
known valuation for the good is denoted by $b_{j}$. We assume that
$s_{i}\in[0,1]$ for all $i\in{\cal S}$ as well as $b_{j}\in[0,1]$ for all
$j\in{\cal B}$. Let $s=(s_{1},\ldots,s_{m})$ and $b=(b_{1},\ldots,b_{n})$. In
addition, let $v=(s,b)$ and $V=[0,1]^{m+n}$. Hence, $V$ is the set of possible
valuation profiles of the sellers and the buyers. We use the usual notation
such as $v=(s_{i},s_{-i},b)=(s,b_{j},b_{-j})$. We also use $v=(v_{i},v_{-i})$
for $i\in{\cal S}$ and $v=(v_{j},v_{-j})$ for $j\in{\cal B}$. That is,
$(v_{i},v_{-i})=(s_{i},s_{-i},b)$ for $i\in{\cal S}$ where $v_{i}=s_{i}$ and
$v_{-i}=(s_{-i},b)$, and $(v_{j},v_{-j})=(s,b_{j},b_{-j})$ for $j\in{\cal B}$
where $v_{j}=b_{j}$ and $v_{-j}=(s,b_{-j})$.

Let $p_{i}:V\rightarrow[0,1]$ be the allocation rule for seller $i\in{\cal
S}$, and let $q_{j}:V\rightarrow[0,1]$ be the allocation rule for buyer
$j\in{\cal B}$. Both $p_{i}(v)$ and $q_{j}(v)$ denote the probability of
trade. In addition, let $x_{i}:V\rightarrow I\\!\\!R$ be the transfer rule for
seller $i\in{\cal S}$ such that $x_{i}(v)$ is the receipt of the money seller
$i$ gets, and let $y_{j}:V\rightarrow I\\!\\!R$ be the transfer rule for buyer
$j\in{\cal B}$ such that $y_{j}(v)$ is the payment of the money buyer $j$
makes. Note that $x_{i}(v)$ may be positive or negative: when it is negative,
seller $i$ pays that amount. Likewise, $y_{j}(v)$ may be positive or negative:
when it is negative, buyer $j$ receives that amount. The payoff of seller
$i\in{\cal S}$ and buyer $j\in{\cal B}$ is denoted respectively by

$u_{i}(v)=x_{i}(v)-p_{i}(v)s_{i}{\rm\ \ \ and\ \ \
}u_{j}(v)=q_{j}(v)b_{j}-y_{j}(v).$

Let $p=(p_{1},\ldots,p_{m}),q=(q_{1},\ldots,q_{n}),x=(x_{1},\ldots,x_{m})$,
and $y=(y_{1},\ldots,y_{n})$. The mechanism $(p,q,x,y)$ is said to be dominant
strategy incentive compatible if

$\eqalign{&u_{i}(s,b)\geq
x_{i}(s^{\prime}_{i},s_{-i},b)-p_{i}(s^{\prime}_{i},s_{-i},b)s_{i}\ \ \forall
i\in{\cal S},\forall s_{i},\forall s^{\prime}_{i},\forall s_{-i},\forall
b;\cr&u_{j}(s,b)\geq
q_{j}(s,b^{\prime}_{j},b_{-j})b_{j}-y_{j}(s,b^{\prime}_{j},b_{-j})\ \ \forall
j\in{\cal B},\forall b_{j},\forall b^{\prime}_{j},\forall b_{-j},\forall s}$
$None$

and ex-post individually rational if

$u_{i}(v)\geq 0\ \ \forall i\in{\cal S}{\rm\ \ and\ \ }u_{j}(v)\geq 0\ \
\forall j\in{\cal B},\ \ \forall v\in V.$ $None$

The mechanism $(p,q,x,y)$ is said to be ex-post budget balancing if

$\sum_{i\in{\cal S}}x_{i}(v)=\sum_{j\in{\cal B}}y_{j}(v)\ \ \forall v\in V$
$None$

and non-wasteful if

$\sum_{i\in{\cal S}}p_{i}(v)=\sum_{j\in{\cal B}}q_{j}(v)\ \ \forall v\in V.$
$None$

Note well that we may have $\sum_{i\in{\cal S}}p_{i}(v)>\sum_{j\in{\cal
B}}q_{j}(v)$ if the mechanism withholds/destroys some units. We exclude this
possibility. We will call the mechanisms that satisfy $(IC)$, $(IR)$, $(BB)$
and $(NW)$ as robust double auction mechanisms. We first have the following
standard lemma.

Lemma 1. The mechanism $(p,q,x,y)$ is dominant strategy incentive compatible
(IC) if and only if

(i) $\forall i\in{\cal S},\forall v_{-i}$: $p_{i}(v)$ is weakly decreasing in
$v_{i}$.

(ii) $\forall j\in{\cal B},\forall v_{-j}$: $q_{j}(v)$ is weakly increasing in
$v_{j}$.

(iii) $\forall i\in{\cal S},\forall v$:
$u_{i}(v)=\int_{v_{i}}^{1}p_{i}(w,v_{-i})dw+u_{i}(1,v_{-i})$.

(iv) $\forall j\in{\cal B},\forall v$:
$u_{j}(v)=\int_{0}^{v_{j}}q_{j}(w,v_{-j})dw+u_{j}(0,v_{-j})$.

Proof: Omitted since it is standard. See Myerson (1981), Myerson and
Satterthwaite (1983), etc. Q.E.D.

We will restrict our attention to the deterministic allocation rules that take
only the value zero or one. Thus, $p_{i}:V\rightarrow\\{0,1\\}$ for all
$i\in{\cal S}$ and $q_{j}:V\rightarrow\\{0,1\\}$ for all $j\in{\cal B}$. Let
$S(v)=\\{i\in{\cal S}|p_{i}(v)=1\\}$ denote the set of trading sellers at $v$,
and let $B(v)=\\{j\in{\cal B}|q_{j}(v)=1\\}$ denote the set of trading buyers
at $v$.

A possible property that the mechanism may satisfy is that all sellers face
one identical price and all buyers face another identical price.

Assumption 1. (Common price) $\forall v\in
V,\forall\\{i,i^{\prime}\\}\subseteq S(v),\forall\\{j,j^{\prime}\\}\subseteq
B(v)$:

$x_{i}(v)=x_{i^{\prime}}(v){\rm\ and\ }y_{j}(v)=y_{j^{\prime}}(v).$

Hence, there are two prices, the seller price (ask price) of $\pi^{s}(v)$ and
the buyer price (bid price) of $\pi^{b}(v)$. Of course, these two prices may
be the same. Note that $\pi^{s}(v)$ and $\pi^{b}(v)$ may vary with $v$. This
assumption is reasonable since all units of the good are identical and the
players supply or demand at most one unit. Observe in particular that, though
the famous VCG (Vickrey-Clarke-Groves) mechanisms may in general induce
different transfers of money across players, there is only one seller price
when each seller supplies one unit and only one buyer price when each buyer
demands at most one unit. In fact, this assumption seems to be a prerequisite
for any centralized trading institution. Example 1 below shows that we may
pair one seller and one buyer and let the trade occur only between them. Then,
each seller-buyer pair is a separated market and different prices across the
pairs may emerge. Although we do not explicitly consider decentralized trading
or resale possibilities in the current mechanism design approach, if the
buyers face different prices, say, then it is conceivable that a buyer of the
more expensive unit could find a new trade opportunity with a seller to the
advantage of both.

Another possible property that the mechanism may satisfy is that the payoff of
a player with the worst possible valuation is equal to zero: Observe that
$u_{i}(v)$ is lowest when $v_{i}=1$ and $u_{j}(v)$ is lowest when $v_{j}=0$ by
Lemma 1(iii)-(iv).

Assumption 2. (Zero payoff for the worst type)

$\forall i\in{\cal S},\forall v_{-i}:u_{i}(1,v_{-i})=0$ and $\forall j\in{\cal
B},\forall v_{-j}:u_{j}(0,v_{-j})=0$.

Note that the condition $u_{j}(0,v_{-j})=0$ for the buyer $j\in{\cal B}$ holds
if and only if $y_{j}(0,v_{-j})=0$ holds. The latter condition is often
assumed in the literature. Observe that $(IR)$ implies $y_{j}(0,v_{-j})\leq 0$
whereas the no subsidy condition implies $y_{j}(0,v_{-j})\geq 0$, leading us
to $y_{j}(0,v_{-j})=0$ and hence $u_{j}(0,v_{-j})=0$.5 5 The no subsidy
condition is (i) $\forall j\in{\cal B},\forall v:y_{j}(v)\geq 0$, and (ii)
$\forall i\in{\cal S},\forall v:x_{i}(v)\leq p_{i}(v)$. The seller case is
essentially symmetric. Observe that $(IR)$ implies $x_{i}(1,v_{-i})\geq
p_{i}(1,v_{-i})$ whereas the no subsidy condition implies $x_{i}(1,v_{-i})\leq
p_{i}(1,v_{-i})$, leading us to $u_{i}(1,v_{-i})=0$.

We now characterize some necessary conditions that a robust double auction
mechanism must satisfy. We use the notation $|A|$ to denote the cardinality of
an arbitrary set $A$, and the notation $v_{-T}$ to denote the valuation
profile of the players not in the set $T\subseteq{\cal S}\cup{\cal B}$.

Proposition 1. Let the mechanism $(p,q,x,y)$ be a robust double auction
mechanism. Under Assumption 1 of common price and Assumption 2 of zero payoff
for the worst type, the mechanism $(p,q,x,y)$ must be the following form: for
all $v\in V$ and $T(v)=S(v)\cup B(v)$,

(i) $|S(v)|=|B(v)|$;

(ii) $x_{i}(v)=y_{j}(v)=\pi(v_{-T(v)})$ for all $i\in S(v)$ and for all $j\in
B(v)$;

(iii) $x_{i}(v)=y_{j}(v)=0$ for all $i\notin S(v)$ and for all $j\notin B(v)$;

(iv) $v_{i}\leq\pi(v_{-T(v)})$ for all $i\in S(v)$ and
$v_{j}\geq\pi(v_{-T(v)})$ for all $j\in B(v)$.

Remark: Property (i) says that the number of trading sellers must be equal to
the number of trading buyers. Property (ii) says that all trading sellers and
trading buyers face the same price $\pi(v_{-T(v)})$ that is independent of
their valuations. Note in particular that there is only one price, i.e., the
seller price is identical to the buyer price. Property (iii) says that the
sellers and the buyers who do not trade neither receive nor pay any money.
Property (iv) says that every trading seller’s valuation is less than or equal
to the price and every trading buyer’s valuation is greater than or equal to
the price.

Proof: It is obvious that $(NW)$ implies property (i). Next, since
$p_{i}(v_{i},v_{-i})\in\\{0,1\\}$, Lemma 1(i) implies: for a given $v_{-i}$,
there exists $z_{i}(v_{-i})\in[0,1]$ such that

$p_{i}(v_{i},v_{-i})=\cases{1&if $v_{i}<z_{i}(v_{-i})$;\cr 0&if
$v_{i}>z_{i}(v_{-i})$.}$

Likewise, Lemma 1(ii) implies: for a given $v_{-j}$, there exists
$z_{j}(v_{-j})\in[0,1]$ such that

$q_{j}(v_{j},v_{-j})=\cases{0&if $v_{j}<z_{j}(v_{-j})$;\cr 1&if
$v_{j}>z_{j}(v_{-j})$.}$

That is,

$\eqalign{&z_{i}(v_{-i})=\sup\\{v_{i}\in[0,1]|p_{i}(v_{i},v_{-i})=1\\}{\rm\ \
for\ }i\in{\cal
S};\cr&z_{j}(v_{-j})=\inf\\{v_{j}\in[0,1]|q_{j}(v_{j},v_{-j})=1\\}{\rm\ \ for\
}j\in{\cal B}.}$

Then, by Lemma 1(iii) and Assumption 2,

$u_{i}(v)=\cases{z_{i}(v_{-i})-v_{i}&when $p_{i}(v_{i},v_{-i})=1$;\cr 0&when
$p_{i}(v_{i},v_{-i})=0$.}$

Likewise, by Lemma 1(iv) and Assumption 2,

$u_{j}(v)=\cases{v_{j}-z_{j}(v_{-j})&when $q_{j}(v_{j},v_{-j})=1$;\cr 0&when
$q_{j}(v_{j},v_{-j})=0$.}$

Hence, we have:

(i) For a seller $i\in{\cal S}$:

$x_{i}(v)=\cases{z_{i}(v_{-i})&when $p_{i}(v)=1$;\cr 0&otherwise.}$

(ii) For a buyer $j\in{\cal B}$:

$y_{j}(v)=\cases{z_{j}(v_{-j})&when $q_{j}(v)=1$;\cr 0&otherwise.}$

We then have

$\sum_{i\in S(v)}x_{i}(v)=\sum_{j\in B(v)}y_{j}(v)\ \ \forall v\in V$

by $(BB)$ and the fact that $x_{i}(v)=0$ for all $i\notin S(v)$ and
$y_{j}(v)=0$ for all $j\notin B(v)$. Since $\sum_{i\in
S(v)}x_{i}(v)=|S(v)|\pi^{s}(v)$ and $\sum_{j\in
B(v)}y_{j}(v)=|B(v)|\pi^{b}(v)$ by Assumption 1 and $|S(v)|=|B(v)|$ by (i), we
have $\pi^{s}(v)=\pi^{b}(v)$. Let us denote this price by $\pi(v)$.

If $T(v)\neq\emptyset$, then $\pi(v)=z_{k}(v_{-k})$ for all $k\in T(v)$.
Hence, $\pi(v)$ does not depend on $v_{k}$ for any $k\in T(v)$. So, we can
write this price as $\pi(v_{-T(v)})$ for all $v\in V$ and $T(v)\subseteq{\cal
S}\cup{\cal B}$. This proves that properties (ii) and (iii) hold. It is now
clear that $(IR)$ implies property (iv). Q.E.D.

Both Assumption 1 and Assumption 2 are essential for Proposition 1 as the
following examples demonstrate. Note that Assumption 2 is satisfied in Example
1 whereas Assumption 1 is (trivially) satisfied in Example 2.

Example 1. Suppose there are two sellers and two buyers, and consider a
mechanism in which (i) two distinct constant prices $\pi^{s}_{1}=\pi^{b}_{1}$
and $\pi^{s}_{2}=\pi^{b}_{2}$ are posted in advance, and (ii) a trade occurs
between seller 1 and buyer 1 when and only when
$s_{1}<\pi^{s}_{1}=\pi^{b}_{1}<b_{1}$ and a trade occurs between seller 2 and
buyer 2 when and only when $s_{2}<\pi^{s}_{2}=\pi^{b}_{2}<b_{2}$. Hence,
seller 1 and buyer 1 face the posted price $\pi^{s}_{1}=\pi^{b}_{1}$ and trade
only between them, and likewise for seller 2 and buyer 2. This mechanism is a
robust double auction mechanism that violates Assumption 1. In this example,
there are two posted price mechanisms separately applied to two pairs of
traders, the pair of seller 1 and buyer 1 and the pair of seller 2 and buyer
2.

Example 2. Suppose there are one seller and two buyers, and consider a
mechanism in which (i) two prices, the seller price $\pi^{s}$ and the buyer
price $\pi^{b}$, are posted in advance with $\pi^{s}<\pi^{b}$, (ii) a trade
occurs between seller 1 and buyer 1 when and only when
$s_{1}<\pi^{s}<\pi^{b}<b_{1}$, (iii) when a trade occurs, seller 1 receives
$\pi^{s}$, buyer 1 pays $\pi^{b}$, and buyer 2 receives $\pi^{b}-\pi^{s}$,
that is, $x_{1}(v)=\pi^{s}$, $y_{1}(v)=\pi^{b}$, and
$y_{2}(v)=\pi^{s}-\pi^{b}$, and (iv) when there is no trade, every player’s
monetary transfer is equal to zero, that is, $x_{1}(v)=y_{1}(v)=y_{2}(v)=0$.
This mechanism is a robust double auction mechanism without one price for all
traders. The reason Proposition 1 does not hold in this example is that
Assumption 2 is not satisfied.6 6 We note that we can construct a similar
mechanism for the case when there are more than one seller and more than one
buyer.

Observe that, for bilateral trading with one seller and one buyer, i.e.,
${\cal S}={\cal B}=\\{1\\}$, we always have $T(v)={\cal S}\cup{\cal B}$ unless
$T(v)=\emptyset$. Hence, $\pi(v)$ does not depend on either $s_{1}$ or $b_{1}$
whenever $T(v)\neq\emptyset$. This mechanism can be implemented by a posted
price mechanism: A constant price $\pi$ is posted in advance and the trade
occurs when and only when $s_{1}<\pi<b_{1}$.7 7 We assume that players decide
not to trade when they are indifferent between trade and no trade. We follow
this convention throughout the paper. Since it is straightforward to show that
Assumptions 1 and 2 are satisfied for bilateral trading environments,
Proposition 1 implies one of Hagerty and Rogerson’s (1987) results that a
robust double auction mechanism for bilateral trading environments is a posted
price mechanism.

Corollary . A double auction mechanism for bilateral trading environments is
robust if and only if it is a posted price mechanism.

Proof: Observe that Assumption 1 is trivially satisfied for bilateral trading
environments. Observe next that $x_{1}(s_{1},b_{1})=y_{1}(s_{1},b_{1})$ by
$(BB)$ and $p_{1}(s_{1},b_{1})=q_{1}(s_{1},b_{1})$ by $(NW)$. Hence, the total
surplus, i.e., the sum of players’ payoffs, is
$p_{1}(s_{1},b_{1})(b_{1}-s_{1})$. This becomes (i)
$p_{1}(1,b_{1})(b_{1}-1)\leq 0$ when $s_{1}=1$, and (ii)
$-p_{1}(s_{1},0)s_{1}\leq 0$ when $b_{1}=0$. Since the total surplus is
nonpositive when either $s_{1}=1$ or $b_{1}=0$ whereas each player’s payoff is
nonnegative by $(IR)$, the only possibility is that each player gets a zero
payoff. Thus, Assumption 2 is always satisfied for bilateral trading
environments. Proposition 1 then implies that a mechanism in these
environments is robust only if it is a posted price mechanism. The converse
also holds since a posted price mechanism is clearly robust. Q.E.D.

In contrast, a robust double auction mechanism may not be a posted price
mechanism when there are more than two players, as the following example
demonstrates.

Example 3. Suppose there are one seller and two buyers, and consider a
mechanism in which (i) players first submit their valuations (which may or may
not be true valuations), (ii) the price is set to the lower of the buyers’
reported valuations, and (iii) a trade occurs between the seller and the buyer
of the higher valuation when and only when the seller’s valuation is less than
the price. When $s_{1}=0.3$, $b_{1}=0.7$, and $b_{2}\in(0.3,0.7)$, for
instance, the trade occurs and the price $\pi=b_{2}$ depends on buyer 2’s
valuation.

Observe in particular that the mechanism in this example is dominant strategy
incentive compatible $(IC)$, mainly due to the fact that the traders cannot
affect the terms of trade. On the other hand, the non-trading buyer affects
the terms of trade without altering her own allocation or payment. We now
introduce another property often used in the literature to obtain a sharper
result. The mechanism $(p,q,x,y)$ is said to be non-bossy if

(i) For all $i\in{\cal S}$ and for all $v_{i},v^{\prime}_{i}$, and $v_{-i}$:
if $p_{i}(v_{i},v_{-i})=p_{i}(v^{\prime}_{i},v_{-i})$ and
$x_{i}(v_{i},v_{-i})=x_{i}(v^{\prime}_{i},v_{-i})$, then
$x(v_{i},v_{-i})=x(v^{\prime}_{i},v_{-i})$ and
$y(v_{i},v_{-i})=y(v^{\prime}_{i},v_{-i})$.

(ii) For all $j\in{\cal B}$ and for all $v_{j},v^{\prime}_{j}$, and $v_{-j}$:
if $q_{j}(v_{j},v_{-j})=q_{j}(v^{\prime}_{j},v_{-j})$ and
$y_{j}(v_{j},v_{-j})=y_{j}(v^{\prime}_{j},v_{-j})$, then
$x(v_{j},v_{-j})=x(v^{\prime}_{j},v_{-j})$ and
$y(v_{j},v_{-j})=y(v^{\prime}_{j},v_{-j})$.

In words, a mechanism is non-bossy if no player can change others’ payments
without changing his/her allocation or payment. We note that this definition
is somewhat different from the standard one first introduced by Satterthwaite
and Sonnenschein (1981). The standard non-bossiness property is stated as
follows in the current setup: the mechanism $(p,q,x,y)$ is non-bossy if (i)
for all $i\in{\cal S}$ and for all $v_{i},v^{\prime}_{i}$, and $v_{-i}$: if
$p_{i}(v_{i},v_{-i})=p_{i}(v^{\prime}_{i},v_{-i})$ and
$x_{i}(v_{i},v_{-i})=x_{i}(v^{\prime}_{i},v_{-i})$, then
$p(v_{i},v_{-i})=p(v^{\prime}_{i},v_{-i})$,
$q(v_{i},v_{-i})=q(v^{\prime}_{i},v_{-i})$,
$x(v_{i},v_{-i})=x(v^{\prime}_{i},v_{-i})$ and
$y(v_{i},v_{-i})=y(v^{\prime}_{i},v_{-i})$; (ii) for all $j\in{\cal B}$ and
for all $v_{j},v^{\prime}_{j}$, and $v_{-j}$: if
$q_{j}(v_{j},v_{-j})=q_{j}(v^{\prime}_{j},v_{-j})$ and
$y_{j}(v_{j},v_{-j})=y_{j}(v^{\prime}_{j},v_{-j})$, then
$p(v_{j},v_{-j})=p(v^{\prime}_{j},v_{-j})$,
$q(v_{j},v_{-j})=q(v^{\prime}_{j},v_{-j})$,
$x(v_{j},v_{-j})=x(v^{\prime}_{j},$ $v_{-j})$ and
$y(v_{j},v_{-j})=y(v^{\prime}_{j},v_{-j})$. That is, the conclusion part
includes the statements about allocations as well as payments.8 8 See Thomson
(2016) for a thorough discussion of non-bossiness. Hence, our definition is
weaker than the standard definition.

Assumption 3. (Non-bossiness) The mechanism $(p,q,x,y)$ is non-bossy.

The non-bossiness assumption is an extra property imposed on the mechanism on
top of the properties required by Proposition 1 under which the generalized
posted price mechanism results. We note that this assumption is also imposed
(along with additional assumptions) in the works of Barberà and Jackson (1995)
and Miyagawa (2001) cited in the introduction. We have:

Proposition 2. Let the mechanism $(p,q,x,y)$ be a robust double auction
mechanism. Under Assumption 1 of common price, Assumption 2 of zero payoff for
the worst type and Assumption 3 of non-bossiness, the mechanism $(p,q,x,y)$
must be the following form: for all $v\in V$ and $T(v)=S(v)\cup B(v)$, there
exists a constant price $\pi$ such that

(i) $|S(v)|=|B(v)|$;

(ii) $x_{i}(v)=y_{j}(v)=\pi$ for all $i\in S(v)$ and for all $j\in B(v)$;

(iii) $x_{i}(v)=y_{j}(v)=0$ for all $i\notin S(v)$ and for all $j\notin B(v)$;

(iv) $v_{i}\leq\pi$ for all $i\in S(v)$ and $v_{j}\geq\pi$ for all $j\in
B(v).$

Proof: We know by Proposition 1(ii) that the price does not depend on the
valuation profile of the traders, i.e., $\pi(v)=\pi(v_{-T(v)})$ for all $v\in
V$. We now show that it does not depend on the valuation profile of the non-
traders, either. Let there be a non-trading seller $i\in
S^{c}(v_{i},v_{-i})\cap S^{c}(v^{\prime}_{i},v_{-i})$. This means
$p_{i}(v_{i},v_{-i})=0=p_{i}(v^{\prime}_{i},v_{-i})$, and
$x_{i}(v_{i},v_{-i})=0=x_{i}(v^{\prime}_{i},v_{-i})$ by Proposition 1(iii). We
have
$\pi(v_{i},v_{-T(v)\setminus\\{i\\}})=\pi(v_{i},v_{-i})=\pi(v^{\prime}_{i},v_{-i})=\pi(v^{\prime}_{i},v_{-T(v)\setminus\\{i\\}})$
where the second inequality holds by Assumption 3. The argument for a non-
trading buyer is similar. Q.E.D.

Hence, the set $T(v)=S(v)\cup B(v)$ of traders may vary as the valuation
profile $v$ changes, but the price does not vary for given a set of traders.
We note that this constant price may be different for different set of
traders, as the following example demonstrates.

Example 4. Suppose there are one seller and two buyers. Let $\pi$ and
$\pi^{\prime}$ be two constant prices with $\pi>\pi^{\prime}$. Let the
mechanism be such that

$\eqalign{&(q_{1}(v),y_{1}(v))=\cases{(1,\pi)&if
$s_{1}<\pi<b_{1}$;\cr(0,0)&otherwise,}\cr&(q_{2}(v),y_{2}(v))=\cases{(1,\pi^{\prime})&if
$s_{1}<\pi^{\prime}<b_{2}$ but not
$s_{1}<\pi<b_{1}$;\cr(0,0)&otherwise,}\cr&(p_{1}(v),x_{1}(v))=(q_{1}(v)+q_{2}(v),y_{1}(v)+y_{2}(v)).}$

It is straightforward to show that this mechanism satisfies all properties in
Proposition 2.

Since both the set of sellers and the set of buyers are finite, the family of
the sets of traders is also finite. This leads us to the following definition.

Definition .  A mechanism is a generalized posted price mechanism if, for each
set $T=S\cup B$ of traders with $|S|=|B|$, there is a constant price $\pi$
such that all sellers in $S$ receive $\pi$, all buyers in $B$ pay $\pi$, and
all other players receive or pay nothing.

Proposition 2 thus states that the price is posted for each set of traders in
any robust double auction mechanism that also satisfies Assumptions 1, 2, and
3. Restating it, we have:

Theorem . A robust double auction mechanism that also satisfies Assumption 1
of common price, Assumption 2 of zero payoff for the worst type and Assumption
3 of non-bossiness must be a generalized posted price mechanism.

We note that the converse of the theorem is not true. To see this, consider
Example 4 again but with $\pi<\pi^{\prime}$. The seller with $s_{1}<\pi$ has
an incentive to submit $s^{\prime}_{1}$ with $\pi\leq
s^{\prime}_{1}<\pi^{\prime}$ and trade with buyer 2 when $\pi<b_{1}$ and
$\pi^{\prime}<b_{2}$. Thus, this generalized posted price mechanism is not
dominant strategy incentive compatible. We also note that the theorem is a
statement about the terms of trade, but not about the selection of traders.
There may exist many alternative manners of determining the set $T(v)=S(v)\cup
B(v)$ of traders for each possible $v$, and it appears to be a difficult task
to provide a clear characterization.

3 The linear price mechanism

Are there robust double auction mechanisms that satisfy Assumptions 1 and 2
but not Assumption 3? That is, are there robust double auction mechanisms
other than the generalized posted-price mechanisms? We already know by Example
3 that there are such mechanisms. In this section, we further pursue this
question. We introduce a canonical double auction mechanism, the linear price
mechanism, and analyze it to see what might happen when we dispense with the
non-bossiness property.

Let us first consider the following variant of Example 3. Suppose there are
one seller and two buyers, and consider a mechanism in which (i) the price is
set to buyer 2’s reported valuation $b_{2}$, and (ii) a trade occurs between
seller 1 and buyer 1 when and only when $s_{1}<b_{2}<b_{1}$. In this example,
the only possible trade is between seller 1 and buyer 1, and buyer 2
contributes to the determination of the price. It is easy to see that this is
a robust double auction mechanism. However, this mechanism has the undesirable
feature that buyer 2 never has an opportunity to trade even when her valuation
is high.

This leads us to consider those mechanisms that respect all players’
valuations, which we will call the value-respecting mechanisms. If the
selection of traders depends on players’ valuations, then it must be the case
that the sellers are selected in an increasing of valuations and the buyers
are selected in a decreasing order of valuations. That is, sellers with lower
valuations and buyers with higher valuations are given priorities. Otherwise,
a seller (buyer) who does not trade but has a lower (higher) valuation than a
seller (buyer) who trades may have an incentive to bid higher (lower) to match
the latter’s valuation, destroying incentive compatibility. Moreover, a robust
double auction mechanism satisfying Assumptions 1 and 2 (but not Assumption 3)
must have the necessary properties derived in Proposition 1. In particular,
the price does not depend on the players’ valuations in $T(v)=S(v)\cup B(v)$.
The following mechanism is a natural mechanism with those properties.

Players first submit their respective valuations in this mechanism, which may
or may not be true valuations. Let us order sellers’ submitted valuations in
an increasing order and buyers’ submitted valuations in a decreasing order.
Thus, we have the order statistics $s_{(1)}\leq s_{(2)}\leq\cdots\leq s_{(m)}$
of sellers’ valuations and the order statistics $b_{(1)}\geq
b_{(2)}\geq\cdots\geq b_{(n)}$ of buyers’ valuations. Ties in the valuations
can be broken in any predetermined way. For convenience, we follow the
convention that $s_{(0)}=b_{(n+1)}=0$ and $s_{(m+1)}=b_{(0)}=1$. Let $\kappa$
be the number such that $s_{(\kappa)}\leq b_{(\kappa)}$ and
$s_{(\kappa+1)}>b_{(\kappa+1)}$. That is, $\kappa=\max\\{k|s_{(k)}\leq
b_{(k)}\\}$. Let us call $\kappa$ as the tentative volume of trade. When
$\kappa<\min\\{m,n\\}$, the posted price $\pi$ is set to

$\pi=c_{1}s_{(\kappa+1)}+c_{2}s_{(\kappa+2)}+\cdots+c_{m-\kappa}s_{(m)}+d_{1}b_{(\kappa+1)}+d_{2}b_{(\kappa+2)}+\cdots+d_{n-\kappa}b_{(n)}$

for given nonnegative numbers $c_{1},\ldots,c_{m},d_{1},\ldots,d_{n}$.9 9 Any
pre-specified rule for the case when $\kappa=\min\\{m,n\\}$ is fine for our
purpose, so we do not specify it. We assume these coefficients are such that
$0\leq\pi\leq 1$ holds for any realization of valuations, since the
possibility of trade is precluded otherwise. In particular, since all
valuations belong to the interval $[0,1]$, any convex combination of non-
traders’ valuations will induce $\pi$ between zero and one. Let
$\hat{S}(v)=\\{i\in{\cal S}|s_{i}\leq s_{(\kappa)}$ and $s_{i}<\pi\\}$ and
$\hat{B}(v)=\\{j\in{\cal B}|\ b_{j}\geq b_{(\kappa)}$ and $b_{j}>\pi\\}$. If
$|\hat{S}(v)|=|\hat{B}(v)|$, then all traders in the set $\hat{S}(v)$ and
$\hat{B}(v)$ trade at price $\pi$. Otherwise, rationing is needed. Let
$\sigma:{\cal S}\rightarrow{\cal S}$ be a permutation of sellers such that
seller $\sigma(1)$ is ordered first, seller $\sigma(2)$ is ordered second, and
so on. Likewise, let $\beta:{\cal B}\rightarrow{\cal B}$ be a permutation of
buyers such that buyer $\beta(1)$ is ordered first, buyer $\beta(2)$ is
ordered second, and so on. If $|\hat{S}(v)|>|\hat{B}(v)|$, then $|\hat{B}(v)|$
sellers in $\hat{S}(v)$ are selected in the order of $\sigma:\cal
S\rightarrow\cal S$ to trade with the buyers in $\hat{B}(v)$; if
$|\hat{S}(v)|<|\hat{B}(v)|$, then $|\hat{S}(v)|$ buyers in $\hat{B}(v)$ are
selected in the order of $\beta:\cal B\rightarrow\cal B$ to trade with the
sellers in $\hat{S}(v)$.10 10 If we allow probabilistic allocation rules, then
random rationing is the most prominent rationing rule both in reality and in
the literature: If $|\hat{S}(v)|>|\hat{B}(v)|$, then $|\hat{B}(v)|$ sellers in
$\hat{S}(v)$ are selected randomly to trade with the buyers in $\hat{B}(v)$;
if $|\hat{S}(v)|<|\hat{B}(v)|$, then $|\hat{S}(v)|$ buyers in $\hat{B}(v)$ are
selected randomly to trade with the sellers in $\hat{S}(v)$. It is
straightforward to see that Proposition 3 below can be extended with minor
modification to the random rationing rule.

Let us call this mechanism as the linear price mechanism. Observe that (i) the
tentative volume of trade is set to maximize potential gains from trade, (ii)
the price is a linear combination of non-traders’ valuations, (iii) a seller
(a buyer, respectively) may not trade even when $s_{i}\leq s_{(\kappa)}$
($b_{j}\geq b_{(\kappa)}$, respectively) depending on the level of the price
$\pi$ actually realized, (iv) the actual volume of trade is less than or equal
to the tentative volume of trade $\kappa$, and (v) the players are obliged to
follow the trading decision after the submission of valuations. The following
example illustrates how a linear price mechanism works.

Example 5. The price $\pi$ is set to $b_{(\kappa+1)}$. That is, $d_{1}=1$ and
$c_{1}=\cdots=c_{m}=d_{2}=\cdots=d_{n}=0$. There are 4 sellers and 4 buyers.
Let both $\sigma:\cal S\rightarrow\cal S$ and $\beta:\cal B\rightarrow\cal B$
be identity mappings. When $s_{1}=0.1,s_{2}=0.3,s_{3}=0.5$, $s_{4}=0.7$,
$b_{1}=0.9,b_{2}=0.8,b_{3}=0.6$ and $b_{4}=0.4$, we have $\kappa=3$,
$\pi=0.4$, $\hat{S}(v)=\\{1,2\\}$ and $\hat{B}(v)=\\{1,2,3\\}$. Since
$|\hat{S}(v)|<|\hat{B}(v)|$, the buyer side has to be rationed. Hence, both
sellers 1 and 2 and buyers 1 and 2 trade. Note that both seller 3 and buyer 3
do not trade in spite of $\kappa=3$: Seller 3 does not trade since the price
$\pi$ is lower than his valuation whereas buyer 3 does not trade since she is
ordered after buyers 1 and 2 in the rationing rule $\beta$.

We now show that, when there are at least two sellers as well as at least two
buyers, this mechanism does not satisfy $(IC)$. We start with the following
lemma.

Lemma 2. Let $\min\\{m,n\\}\geq 2$. If a linear price mechanism satisfies
$(IC)$, then

$c_{1}=c_{2}=\cdots=c_{m}=0.$

Proof: We first show that $c_{1}=0$. Suppose to the effect of contradiction
that $c_{1}>0$. Let the realized valuations be such that
$s_{(\kappa)}<b_{(\kappa+1)}<s_{(\kappa+1)}<c_{1}s_{(\kappa+2)}$ and $b_{(1)}$
is sufficiently high.11 11 It helps to think of the case when $\kappa=1$.
Then, seller $i$ with $s_{i}=s_{(\kappa+1)}$ has an incentive to bid
arbitrarily close to 0 when $\sigma(1)=i$. In that case, the new tentative
volume of trade becomes $\kappa+1$ and the new price becomes $\pi^{\prime}\geq
c_{1}s_{(\kappa+2)}$, which is higher than $s_{(\kappa+1)}$. Since there
exists at least one buyer whose valuation $b_{(1)}$ is higher than
$\pi^{\prime}$, seller $i$ gets a positive payoff instead of a payoff of zero
from reporting truthfully.

We next show that $c_{l}=0$ for all $l=2,\ldots,m$. Suppose to the effect of
contradiction that $c_{l}>0$. Let the realized valuations be such that
$b_{(\kappa+1)}<s_{(\kappa)}<s_{(\kappa+1)}<\pi$ and $b_{(1)}$ is sufficiently
high. For instance, let $s_{(\kappa+1)}$ be small enough to satisfy
$s_{(\kappa+1)}<c_{l}s_{(\kappa+l)}$. Then, seller $i$ with
$s_{i}=s_{(\kappa+1)}$ has an incentive to bid arbitrarily close to zero when
$\sigma(1)=i$ since (i) it would increase the probability of trade from zero
to one as well as (ii) it would not change the price since the new price12 12
Observe that the tentative volume of trade does not change. Observe also that,
although $s_{(\kappa)}$ now becomes the $(\kappa+1)$-th lowest valuation of
the seller, the price remains the same since $c_{1}=0$.

$\pi^{\prime}=c_{2}s_{(\kappa+2)}+\cdots+c_{m-\kappa}s_{(m)}+d_{1}b_{(\kappa+1)}+d_{2}b_{(\kappa+2)}+\cdots+d_{n-\kappa}b_{(n)}$

is equal to the original price $\pi$, from which he gets a positive payoff
instead of a payoff of zero from reporting truthfully. Q.E.D.

Hence, we have $\pi=d_{1}b_{(\kappa+1)}+\cdots+d_{n-\kappa}b_{(n)}$. We now
show that this mechanism does not satisfy $(IC)$. Let the realized valuations
be such that

$d_{1}b_{(\kappa+2)}+\cdots+d_{n-\kappa-1}b_{(n)}<b_{(\kappa+1)}<s_{(\kappa+1)}<b_{(\kappa)}$

and $s_{(1)}$ is sufficiently low. Then, buyer $j$ with $b_{j}=b_{(\kappa+1)}$
has an incentive to bid arbitrarily close to 1 when $\beta(1)=j$. In that
case, the new tentative volume of trade becomes $\kappa+1$ and the new price
becomes $\pi^{\prime}=d_{1}b_{(\kappa+2)}+\cdots+d_{n-\kappa-1}b_{(n)}$, which
is lower than $b_{(\kappa+1)}$. Since there exists at least one seller whose
valuation $s_{(1)}$ is lower than $\pi^{\prime}$, buyer $j$ gets a positive
payoff instead of a payoff of zero from reporting truthfully. Summarizing the
discussion, we have:

Proposition 3. The linear price mechanism is not a robust double auction
mechanism when there are more than one seller and more than one buyer.

Hence, even without Assumption 3 of non-bossiness, it seems quite difficult to
find a value-respecting robust double auction mechanism other than the
generalized posted price mechanism.

4 Conclusion

We have shown that the price in any robust double auction mechanism does not
depend on the valuations of the trading sellers or the trading buyers. We have
also shown that, with a non-bossiness assumption, the price in any robust
double auction mechanism does not depend on the players’ valuations at all,
whether trading or non-trading. Our main result is the characterization result
for the general double auction environments that, with a non-bossy assumption
along with other assumptions on the properties of the mechanism, the
generalized posted mechanism is the only robust double auction mechanism.
Furthermore, we have introduced the linear price mechanism to demonstrate
that, even without the non-bossiness assumption, it is quite difficult to find
a reasonable robust double auction mechanism other than the generalized posted
price mechanisms.

Although some of the results can be extended straightforwardly to the class of
probabilistic allocation rules, our main characterization has been established
for deterministic allocation rules. Hence, it remains as a future research
agenda to extend the analysis to probabilistic robust double auction
mechanisms.

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# Kinesthetic-based In-Hand Object Recognition
with an Underactuated Robotic Hand

Julius Arolovitch∗, Osher Azulay∗ and Avishai Sintov ∗ These authors
contributed equally.J. Arolovitch is with the Robotics Institute, Carnegie
Mellon University, Pittsburgh, PA, USA. E-mail<EMAIL_ADDRESS>Azulay and A. Sintov are with the School of Mechanical Engineering, Tel-Aviv
University, Israel. E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Tendon-based underactuated hands are intended to be simple, compliant and
affordable. Often, they are 3D printed and do not include tactile sensors.
Hence, performing in-hand object recognition with direct touch sensing is not
feasible. Adding tactile sensors can complicate the hardware and introduce
extra costs to the robotic hand. Also, the common approach of visual
perception may not be available due to occlusions. In this paper, we explore
whether kinesthetic haptics can provide in-direct information regarding the
geometry of a grasped object during in-hand manipulation with an underactuated
hand. By solely sensing actuator positions and torques over a period of time
during motion, we show that a classifier can recognize an object from a set of
trained ones with a high success rate of almost 95%. In addition, the
implementation of a real-time majority vote during manipulation further
improves recognition. Additionally, a trained classifier is also shown to be
successful in distinguishing between shape categories rather than just
specific objects.

##  I Introduction

Underactuated hands, characterized by their ability to conform to object
shapes through compliance, offer simplicity and cost-effectiveness [1, 2].
This is in contrast to traditional robotic hands, while precise and
proficient, are limited by their intricate design, high costs and complex
control schemes [3, 4]. Underactuated hands have been shown to provide stable
grasps through open-loop control [5] and an ability to perform precise in-hand
manipulation tasks [6, 7, 8]. Some attempts have been made to also enable
object recognition with underactuated hands [9, 10]. These, however, often
require additional sensory hardware on the hand [11, 12, 13].

The recognition of an object grasped within a robotic gripper or hand has been
a topic for numerous works [14, 15]. An ability to recognize object without
visual perception increases the workspace of the robot and enables it to work
in regions without proper lighting or line-of-sight [16]. Relying solely on
continuous visual feedback limits the performance in various tasks where
visual uncertainty or occlusion may occur. In such cases, it may be impossible
to solve the task altogether. Examples include grasping an arbitrary object at
the back of a cabinet or in confined spaces as demonstrated in Figure 1.

Figure 1: In-hand recognition of an object in an occluded environment with an
underactuated hand where visual perception is not available. The hand relies
on kinesthetic perception during in-hand manipulation to either recognize the
specific object from a trained set or its general shape.

The common approach for object recognition is the use of haptics or, in
particular, tactile sensing [17, 18]. In such sensing, tactile sensors have
direct contact with the surface of the object and provide a glance of its
geometry for recognition inference [19]. Often, the hand requires multiple
tactile glances in order to provide a certain recognition regarding the object
[20, 21]. However, adding tactile sensors to robotic hands can be expensive
and may complicate the hardware. Kinesthetic haptics, on the other hand,
gather information by sensing movement, force and position of actuator joints
in the hand [22]. Prior work has combined tactile images along with
kinesthetic information while utilizing $k$-means clustering to increase
classification success rate [23]. More recently, kinesthetic haptics was used
to recognize objects by observing finger kinematics during multiple grasps
[24]. However, these assume rigid hands and full knowledge of the kinematic
configuration of the fingers.

Object recognition using underactuated hands has mostly relied on either
visual methods or tactile perception [25, 26, 27, 28, 29]. In [30], embedded
force sensors along the two fingers of an underactuated hand were introduced
for object classification and feature extraction through single-grasp
interactions. Similarly, optical tactile sensors, which observe deformation on
the contact pad through an internal camera, were used on a three-finger
underactuated hand for recognition of grasped objects [31]. Integrating these
sensors into compliant hands adds extra costs and computational complexity to
the design. This is even more crucial for open-source 3D printed underactuated
robotic hands that attempt to provide a simple, low-cost and accessible
solution [32].

In this paper, we address the problem of proprioceptive object recognition
with underactuated hands by exploring it solely through kinesthetic haptics.
Proprioception, or kinaesthesia, offers a unique way to recognize the state of
the hand-object system without relying on conventional visual or tactile cues.
We focus on a tendon-based underactuated hand where actuators at the base of
the hand pull tendons running along the fingers, which also have passive
joints and springs. With no tactile sensors on the fingers and no exact model
for these compliant mechanisms [8], we observe signals from the actuators
during in-hand manipulation of various objects and explore whether they embed
crucial information regarding their shapes. The feature state representation
and data-based modeling are explored.

A single proprioceptive state signal from the hand at some time instant is not
sufficient for representing the shape of a grasped object. Therefore, we
hypothesize that a sequence of states during in-hand manipulation could embed
geometric information such as curvature, flat surface and corners.
Consequently, we collect sequential data during the manipulation of various
objects. With the data, our investigation delves into an extensive analysis of
various classification models, either simple or temporal-based, that capture
the intricate relationships between the hand’s proprioceptive data and the
underlying object geometries. Furthermore, we utilize a simple majority vote
paradigm for increasing the certainty of the recognition in real-time during
manipulation. Finally, the generalization capabilities of the models are
explored, evaluating their performance on generalizing to categories of shapes
rather than specific objects in the training set.

By focusing on an object recognition model without vision or tactile sensing,
we provide a novel capability that aligns with simplicity and cost-
effectiveness in hardware design. Without additional sensors, a low-cost
underactuated robotic hand can pick-up an object and simply manipulate it
between the fingers for a short period of time in order to recognize it.
Hence, by circumventing the limitations of visual and tactile perception, we
offer a pure algorithmic and potentially more robust approach to in-hand
object recognition. While out of the scope of this work, the detection for a
successful grasp at the pick-up phase can also be recognized through
kinesthetic haptics. We note also that the proposed approach can be
complementary to visual perception in order to increase recognition certainty
or replace it entirely if a line-of-sight is not available.

##  II Method

###  II-A Problem Definition and Approach

Consider a two-finger underactuated hand as seen in Figure 2. The hand
comprises of two opposing tendon-based fingers such that a manipulated object
between the fingers performs a planar motion [8]. Each finger consists of two
compliant joints with springs, and a tendon runs along its length. The tendon
is connected to an actuator situated at the base of the hand and the finger
flexes upon pulling of the tendon. The distal links of the fingers are
equipped with high friction pads to prevent slipping. The hand has no tactile
sensors on the fingers and no visual perception of the hand-object system is
available. The hand can only measure kinesthetic features such as actuator
angles and torques.

Figure 2: An underactuated robotic hand (OpenHand Model-O [32]) with two-
fingers. The hand is mostly 3D printed and is tendon-based. Each finger
consists of two passive joints with springs. The tendons run along the length
of the two fingers and are pulled by two actuators at the base of the hand.

Let $\mathbf{x}\in\mathcal{C}$ represent the observable state of the hand
where $\mathcal{C}\subset\mathcal{R}^{n}$ is some $n$-dimensional state space.
Similarly, vector $\mathbf{a}\in\mathcal{U}$ denotes an action exerted on the
hand where $\mathcal{U}\subset\mathbb{R}^{2}$ is the action space. The action
vector $\mathbf{a}$ corresponds to angle changes of the two actuators,
signifying tendon pull or release, over a fixed time step $\Delta t$. Since no
visual perception is used, the true state of the hand and the pose of the
object $\mathbf{T}\in SE(2)$ with respect to the hand are not known and cannot
be observed directly [16].

Given a set of $m$ objects $\\{\mathcal{O}_{1},\ldots\mathcal{O}_{m}\\}$, it
is required to identify an object grasped by the hand from the set without any
use of visual feedback nor tactile sensors. The query object will be
identified during the grasp without releasing and regrasping it. It is assumed
that all grasped objects are rigid. Furthermore, there is no assumption on the
pose $\mathbf{T}_{0}$ of the object during the initial grasp. The main
objective of this work is, therefore, twofold. First, we aim to identify the
set of $n$ kinesthetic features that best represent relevant information
regarding the shape of the grasped object. Second, we wish to utilize the
features and explore the ability of a data-based model to classify grasped
objects. Consider state measurements arriving sequentially
$\\{\mathbf{x}_{1},\ldots,\mathbf{x}_{k}\\}$ in real-time while manipulating a
given query object $\mathcal{O}$ whose class is unknown. It is required to
solve the following maximization problem

$j^{*}=arg\max_{j}P(\mathcal{O}_{j}|\mathbf{x}_{1},\ldots,\mathbf{x}_{k})$ (1)

where $P(\mathcal{O}_{j}|\mathbf{x}_{1},\ldots,\mathbf{x}_{k})$ is the
conditional probability for $\mathcal{O}$ to be in class $\mathcal{O}_{j}$.

###  II-B Data Representation

Data is collected by recording various state transitions during in-hand
manipulation of the $m$ objects. An observable state of the hand during
manipulation is a kinesthetic measurement of internal features. In the tested
hand, the state can consist of actuator torques and angles. In addition, we
consider the case where the applied action at the given time is included in
the state. Hence, the state $\mathbf{x}$ can be of dimension up to $n=6$. The
collection of data is conducted in an episodic manner where, in each episode,
an object is manipulated by exerting a set of pre-defined actions until drop.
Hence, data for episode $i$ is a set of states
$\mathcal{E}_{i}=\\{\mathbf{x}_{0},\ldots,\mathbf{x}_{z}\\}$ and the
corresponding object label $y_{i}\in\\{1,\ldots,m\\}$. In order to overcome
biases throughout different episodic sessions, the initial state of an episode
is subtracted from all states in the episode, i.e.,
$\tilde{\mathbf{x}}_{j}=\mathbf{x}_{j}-\mathbf{x}_{0}$. Subsequently, an
optimized low pass filter is applied to each episode to alleviate sensor
noise. Consequently, dataset $\mathcal{D}$ is of the form

$\mathcal{D}=\\{(\tilde{\mathcal{E}}_{i},y_{i})\\}_{i=1}^{N_{e}}$ (2)

consisting of $N_{e}$ episodes and where
$\tilde{\mathcal{E}}_{i}=\\{\boldsymbol{0},\tilde{\mathbf{x}}_{1},\ldots,\tilde{\mathbf{x}}_{z}\\}$.

###  II-C Data-based Model and Inference

The recognition of a grasped object without tactile sensors cannot be done
with only a single kinesthetic state $\mathbf{x}_{i}$. Such state will not
have sufficient information regarding the shape of the object. Alternatively,
state change during manipulation may have embedded geometric information such
as corners and surface curvature across the object. Hence, we hypothesize that
a sequential set of states $\\{\mathbf{x}_{1},\ldots,\mathbf{x}_{k}\\}$ would
enable the training of a model for the desired recognition.

Sequential data enables to learn the motion pattern of the hand-object system
over some period of time. Given a set of $w$ sequential states along a motion
time frame $S_{w}=\\{\mathbf{x}_{1},\ldots,\mathbf{x}_{w}\\}$, we search for a
data-based classification model that could provide a solution for (1). In
practice, we aim to train a model
$\Gamma_{\psi}:\mathcal{C}\times\ldots\times\mathcal{C}\to[0,1]^{m}$, where
$\psi$ is the vector of trained parameters, to provide a probability
distribution over the $m$ possible objects in the form

$\Gamma_{\psi}(S_{w})=\left[P(\mathcal{O}_{1}|S_{w}),\ldots,P(\mathcal{O}_{m}|S_{w})\right].$
(3)

Then, the predicted object would be $\mathcal{O}_{j}$ with

$j^{*}=arg\max_{j}\Gamma_{\psi}(S_{w})]$ (4)

where $arg\max_{i}\mathbf{v}$ returns the index of the component in vector
$\mathbf{v}$ with the maximum value. In order to train model $\Gamma_{\psi}$,
dataset $\mathcal{D}$ is modified to include labeled motion sequences of pre-
defined length $w$. Along each episode $\tilde{\mathcal{E}}_{i}$ of length
$z$, a window of length $w$ is moved to generate $z-w+1$ sequences
$\\{S_{w,i}^{(1)},\ldots,S_{w,i}^{(z-w+1)}\\}$. Each sequence $S_{w,i}^{(j)}$
is labeled with the original label $y_{i}$ yielding a training set
$\mathcal{P}=\\{(S_{w,i},y_{i})\\}_{i=1}^{N}$.

Training a sequential model with $\mathcal{P}$ for $\Gamma_{\psi}$ in (3) can
be done with data-based architectures designated for sequence or temporal
modeling, such as the Long Short-Term Memory (LSTM) [33] and Temporal
Convolutional Network (TCN) [34]. LSTM is a class of Recurrent Neural-Networks
(RNN) aimed to learn sequential data and is able to selectively retain or
discard information from previous time steps making it well-fit for long-term
dependencies. TCN is a type of neural-network that uses convolutional layers
to process sequential data. The convolutional layers extract significant
features from the data. By using dilated convolutions, the TCN is able to
capture long-term dependencies in a computationally efficient manner, making
it a popular and efficient choice for temporal prediction tasks.

Each of the above models and other classifiers will output a probability
distribution as in (3) and a prediction according to (4). However, the maximum
probability for a specific object class may not be sufficiently high in order
to have a high certainty prediction. Nevertheless, one may exploit the
continuous time frame in which the hand manipulates a certain object and
rapidly acquire additional samples while being certain that they originate
from the same object. Hence, a majority vote can be used to accumulate
probabilities over time [35]. Let
$\mathbf{p}_{t}=\Gamma_{\psi}(S_{w.i}^{(t)})$ be the probability distribution
over the $m$ object at time instance $t$. An accumulated prediction after $T$
time frames is the object $\mathcal{O}_{i^{*}}$ that acquires the maximum sum
of scores given by

$i^{*}=arg\max_{i}\sum_{t=1}^{T}\mathbf{p}_{t}.$ (5)

In this way, the model can keep accumulate and improve its prediction over
time.

##  III Experiments

###  III-A Experimental Setup

An experimental system was constructed based on the two opposing fingers of
the OpenHand Model-O [32] underactuated hand as seen in Figure 2. The
experimental system consists of the hand and an automated reset mechanism.
Once the object drops from the fingers, a thin string that runs through a hole
in the center of the object pulls it into the reach of the fingers toward a
new random grasp. A camera and a fiducial marker on the object are used solely
to confirm the successful initial grasp in each episode. An action is defined
to be
$\mathbf{a}\in\\{0^{\circ},1^{\circ},-1^{\circ}\\}\times\\{0^{\circ},1^{\circ},-1^{\circ}\\}$
where, in each step, the actuators are stalled or moved by a constant amount.
The system is operated by the Robot Operating System (ROS).

Eight objects, seen in Figure 3, are used for training and evaluating the
classification model. Seven of the objects are prismatic PLA ones with
different cross-sections including square, star-like, arbitrary-curved, half-
circular, circular, hexagonal and elliptical. The eighth object is a rubber
duck. Data was collected for the object where, in each episode, the hand
grasps the object, performs an in-hand manipulation with a pre-defined action
sequence for 10 seconds and then drops the object. The hand has no tactile
sensors while only actuator angles and torques can be measured. During
manipulation, data stream of these features is available in 10 Hz and recorded
including instantaneous actions.

Figure 3: Eight distinct objects used for training the classification model. TABLE I: Mean recognition success rate for various feature combinations and models  Feature combination ($n$) | LSTM | TCN | FC-NN | RF | SVM
---|---|---|---|---|---
(%) | (%) | (%) | (%) | (%)
Torques (2) | 38.88 | 52.55 | 49.75 | 62.42 | 48.27
Torques and actions (4) | 65.54 | 76.8 | 73.04 | 74.73 | 69.94
Torques and angles (4) | 90.02 | 94.86 | 93.49 | 93.88 | 82.57
Torq., ang. and act. (6) | 83.06 | 94.03 | 87.64 | 92.35 | 79.66

###  III-B Object Recognition

We begin by evaluating the classification accuracy over the eight objects.
Dataset $\mathcal{D}$ was collected over $N_{e}=5,010$ episodes with
approximately 625 episodes per object. The dataset was processed with the full
length of the sequences, i.e., $w=100$, yielding a training dataset
$\mathcal{P}$ with $N=5,010$ labeled sequences and 501,000 state samples. An
additional test dataset of $860$ labeled sequences was collected and not
included in the training set in any way.

We compare between several classification models including LSTM and TCN. While
these are designated for temporal modeling, we also compare general
classifiers including Random Forest (RF), Support Vector Machine (SVM) and a
fully-connected NN (FC-NN). The hyper-parameters of the models were optimized
yielding the following architectures. The TCN consists of three layers, two
with 256 neurons, one with 128 neurons and a ReLU activation in between. A
dropout of 10% and an L1 regularizer with a factor of $10^{-3}$ were included
to reduce over-fitting. Learning rate scheduling was also utilized, starting
with a learning rate of 0.0001 and terminating at a rate of $10^{-6}$. The
LSTM also consists of three layers consisting of 128, 64 and 32 neurons. A
dropout of 5% and L1 regularizer with a factor of $10^{-2}$ were used to
decrease over-fitting. A learning rate scheduler was also used, initializing
at 0.001 and terminating at a rate of $10^{-6}$. While LSTM and TCN can
receive sequential input, the input for RF, SVM and FC-NN is a single vector.
Hence, each sequence $S_{w,i}$ was flattened to a vector of size $wn$. The
Random Forest consisted of 100 decision trees. The FC-NN consists of two
layers of 128 and 64 neurons. A dropout of 10% and L1 regularizer with a
factor of $10^{-4}$ were used. A learning rate scheduler was also used,
initialized at 0.001 and terminated at a rate of $10^{-6}$.

Table I summarizes the results of the four models over the test data with four
varying feature combinations of actuator angles, torques and actions. First,
it is clear that torque signals along the motion are not sufficient in order
to characterize the shape of the manipulated object. Including the actions
provides marginal improvement since they are rather sparse. However, the best
results are achieved with full state of the actuators having torques and
angles. This four-dimensional state formulation was used in subsequent
testing. When comparing between the five classification models, TCN with
torques-angles state achieves the highest success with marginal advantage
compared to FC-NN and RF. The results show that even simpler classifiers can
successfully recognize object during in-hand manipulation. SVM provides
inferior results to all models. Figure 4 exhibits the confusion matrix for
classifying the eight objects with the TCN model. Results show high
recognition rate for all objects in the set.

Figure 4: Confusion matrix for classification of the eight objects with the
TCN model over the test data.

We next observe performance of the models with regards to the size of the
training data. Recall that the full size of the training data is $N=5,010$
sequences amounting to approximately 16 hours of automated data collection.
For a fair analysis, models were trained on varying portions of the data
where, in each evaluation, data is picked from $\mathcal{D}$ without any
shuffling so to maintain sequential order. For each data size, the model is
re-trained 20 times on different parts of the data and the success rate is
averaged. Results for success rate with regards to portion of the data in
$\mathcal{P}$ used to train the model can be seen in Figure 5. Success rate
saturation is reached for all models with about 60% of the dataset which
corresponds to approximately 10 hours of data collection. TCN reaches the
highest value in saturation whereas RF has higher values with a lower amount
of training data. SVM performed the poorest and is not used in further
evaluations.

Figure 5: Object recognition success rate with regards to the fraction of
total training data used to train RF, LSTM, SVM, FC-NN and TCN.

The above trained models used sequence length $w=100$ for comparison. Analysis
is now presented on the performance with regards to the value of $w$. Here
also, for each value of $w$, the model was re-trained 20 times and the success
rate was averaged. Figure 6 depicts the success rate with regards to the
length of the sequence $w$ over the test data. All models saturate with a
sequence length $w>75$. This result shows that sufficient motion must be
exerted in order to acquire informative data for accurate recognition.

Figure 6: Object recognition success rate with regards to the length of the
sequence $w$ in training and testing models TCN, LSTM, FC-NN and RF.

The use of the majority vote in order to accumulate certainties over objects
and improve success rates is further evaluated. For the majority vote, we use
a sequence size of $w=75$ and slide it over 25 inferences in order to
accumulate probability distributions according to (5). The majority vote is
compared to single inferences with sequences of length $w=75$ and $w=100$.
Table II presents the results over four classification models. The results
show marginal improvement when using the whole episode length of $w=100$ over
a shorter one $w=75$ in a single inference classification. However, exploiting
the entire episode but with a majority vote over sequences of $w=75$ provides
extremely high success rates. Hence, even an ill-trained classifier can be
used in a majority vote setting in order to improve predictions. In general,
it is more advantageous to train a model with shorter sequence length and with
majority vote, rather then having single inference with a longer sequence. The
performance of the majority vote in real-time and with various sequence
lengths will be analyzed later on.

TABLE II: Classification success rate of the eight objects for single inferences and with majority votes |  | LSTM | TCN | RF | FC-NN
---|---|---|---|---|---
|  | (%) | (%) | (%) | (%)
Single inference | $w=75$ | 89.80 | 93.17 | 91.67 | 93.00
Single inference | $w=100$ | 90.02 | 94.86 | 93.88 | 93.49
Majority vote | $w=75$ | 93.60 | 99.90 | 98.99 | 99.67

###  III-C Generalized Recognition

The ability of a model to classify everyday objects into shape categories and
not only specific objects is now evaluated. We consider four shape categories:
circular (7 objects), semi-circular (6 objects), square (9 objects) and
elliptical (7 objects) prisms. A set of household test objects of varying
sizes and within the categories was collected for benchmarking the
generalization ability of the trained models. An example for the set of
circular objects is seen in Figure 7. A successful recognition is the one that
matches the shape category of a test object to the corresponding train object
(seen in Figure 3). Table III compares success rates between single inferences
and majority vote for the shape categories. The classification of entire
sequences ($w=100$) proved to be the least efficient in generalization, less
so than classifying sequences of length $w=75$. The majority vote proves to be
resilient in overcoming shape and size differences. Table IV presents the size
ranges with respect to the train objects that reached a success rate above 75%
and the actual success rates for majority votes of $w=75$ sequences. The
results show a distinctive ability for the trained models to recognize shapes
of new objects not included in the training and with different sizes.

Figure 7: Circular household objects used for testing the generalization ability of the trained classifiers and their radii. TABLE III: Classification success rate of four shape categories for single inferences and with majority votes |  | LSTM | TCN | RF | FC-NN
---|---|---|---|---|---
|  | (%) | (%) | (%) | (%)
Single inference | $w=75$ | 80.63 | 87.76 | 95.50 | 94.42
Single inference | $w=100$ | 76.88 | 87.38 | 91.75 | 90.61
Majority vote | $w=75$ | 89.59 | 94.55 | 98.00 | 96.04
TABLE IV: Size range with respect to the training objects and success rate for recognizing the shape category of new objects with the TCN classifier  Shape category | Size Range (%) | Success rate (%)
---|---|---
Circular | 32.2–193.7 | 91.5
Semi-Circular | 56.0–181.5 | 95.5
Square | 43.5–135.3 | 86.0
Elliptical | 79.3–145.6 | 94.0
Figure 8: Success rate of shape recognition in during the manipulation of
various objects using real-time majority votes. Figure 9: Snapshots of real-
time object recognition of an hexagon prism over 25 time steps and 2.5
seconds. The model certainty of grasping the prism after 80, 90 and 100 time
steps is 0.898, 0.914 and 0.920, respectively. Figure 10: Snapshots of real-
time object recognition of a rubber duck over 25 time steps and 2.5 seconds.
The model certainty of grasping the duck after 80, 90 and 100 time steps is
0.923, 0.961 and 0.969, respectively. Figure 11: Snapshots of real-time object
recognition of a soda can over 25 time steps and 2.5 seconds. The model
certainty of grasping the can after 80, 90 and 100 time steps is 0.726, 0.776
and 0.812, respectively.

###  III-D Real-Time Object Recognition

In the last experiment, we evaluate the ability of the model to conduct live
object recognition early into the manipulation sequence and the prediction
improvement with the majority vote. While we discuss real-time inference, we
simulate real-time inference along the recorded test episodes and evaluate
recognition success rate, focusing particularly on the model’s ability to
conduct generalized real-time recognition having been trained on a set of
generic objects. Since a recorded episode is of length 10 seconds or 100
steps, a majority vote would have $T=100-w$ instances to collect predictions
and make inferences according to (5). Let $S_{w}^{(t)}$ be a sequence of
length $w$ arriving at time $t$. Hence, once the first sequence $S_{w}^{(1)}$
arrives, an initial inference can be computed. Then, as additional states
arrive, more sequences are acquired
$\\{S_{w}^{(2)},S_{w}^{(3)},\ldots,S_{w}^{(T)}\\}$ and predictions can be
improved by updating the majority vote in (5).

Figure 8 presents the success rate analysis with the updating of the majority
vote along time for a series of trials of general shape recognition. The
analysis compares between different lengths of sequences used in inferences
for the majority voting. First, having $w\leq 20$ yields a decline in success
rate. This is due to the significantly ill-trained and faulty classifier
trained over short sequences with insufficient information. The conditions for
a classifier to guarantee prediction improvement in a majority vote are given
in [24]. Classifiers with longer sequences are shown to improve predictions
over time while reaching saturation at some point. Longer sequences, however,
can provide a first prediction later in the episode. Sequences of
approximately $w=75$ provide the highest success rate almost from the first
inference with 88% to saturation at 98%. Figures 9-11 show snapshots of real-
time object recognition inferences with three different objects. The motion
that the object-hand system has to exert is rather small before a high
certainty recognition is acquired. Hence, the hand can quickly move on to
performing the desired task.

##  IV Conclusions

In this work, we have explored the ability of a tendon-based compliant hand to
recognize grasped objects without any tactile sensing nor visual perception.
We have shown that using only kinesthetic haptics during in-hand manipulation
can provide sufficient information regarding the shape of the objects. In
other words, with only position and torque data of the hand’s internal
actuators during motion with the objects, one can train a classifier to
distinguish between the objects. A set of classification models were
benchmarked including temporal ones. The results show best performance using
the TCN with a marginal advantage over simple FC-NN or RF. In addition,
majority vote with constant inferences during the hand-object motion is shown
to increase prediction certainty and success rate. The proposed approach is
also able to recognize a geometry characteristic of an object leading to a
more general recognition model. The findings of this study have the potential
to augment the capabilities of low-cost and 3D printed underactuated hands
without any requirement for additional sensing hardware.

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