Players may be allowed to both buy and sell the same commodity in a strategic market game. Any outcome of such a game can also be obtained as an outcome of a game in which players either buy or sell. However, an equilibrium allocation of the buy and sell game may not be so in the corresponding buy or sell game as the set of achievable allocations for any player, given others' strategies, is different in these two games.Strategic market game; Buy and sell; Buy or Sell.

Indrajit Ray

English

Research Papers in Economics

 
 
 
 
 
 
 
 
 
 
 
 
Discussion Papers in Economics 
 
 
 
 
 
 
 
 
 
 
No. 2000/62 
 
Dynamics of Output Growth, Consumption and Physical Capital 
in Two-Sector Models of Endogenous Growth 
 
by 
 
Farhad Nili 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Department of Economics and Related Studies 
University of York 
Heslington 
York, YO10 5DD 
 
No. 2000/13 
 
Buying and Selling in Strategic Market Games 
 
by 
 
Indrajit Ray 
      
*I would like to thank Gianni De Fraja and the participants of the Royal Economic Society Conference at Nottingham for
many helpful comments.
     
**Department of Economics, University of York, Heslington, York YO10 5DD, UK.
1
Buying and Selling in Strategic Market Games
*
Indrajit Ray
**
November 1999 2
Buying and Selling in Markets
Indrajit Ray
Department of Economics
University of York
Heslington, York YO10 5DD
United Kingdom
E-mail : ir2@york.ac.uk
Fax : 44-1904-433759
Abstract
Players may be allowed to both buy and sell the same commodity in a strategic market game. Any
outcome of such a game can also be obtained as an outcome of a game in which players either buy or
sell. However, an equilibrium allocation of the buy and sell game may not be so in the corresponding
buy or sell game as the set of achievable allocations for any player, given others’ strategies, is different
in these two games.
Journal of Economic Literature Classification Number: C72.
Key Words : Strategic market game, Buy and sell, Buy or Sell.3
1. INTRODUCTION
Shapley and Shubik (Shapley, 1976; Shapley and Shubik, 1977) introduced and developed
strategic market games as a method of making a non-cooperative game out of an exchange economy.
Strategic market games are quite popular among theorists as it offers a model to study game theoretic
analogs of well-known results in economic theory, particularly, general equilibrium theory. In a market
game, players are allowed to both buy and sell the same commodity. Such a buy and sell mechanism
has mostly been used in the literature. Buy and sell mechanism allows each player a lot more strategic
freedom; however, it is intuitively difficult to justify why agents would sell and buy back the same
commodity from the market. There are other variations of strategic market game models : buy or sell,
and sell all. Buy or sell mechanism is clearly an appealing mechanism; an agent in such a market either
buys a commodity or sells it off.
The purpose of the paper is to understand precisely the difference or the equivalence, if any,
between buy and sell, buy or sell mechanisms. We therefore concentrate on the set of (equilibrium)
outcomes of a buy and sell game and that of the corresponding buy or sell game. We first prove that any
outcome of a buy and sell game can be obtained as an outcome of the corresponding buy or sell game.
Thus these two games are payoff equivalent (Ray, 1999). 
Two payoff equivalent games may not have the same set of equilibria. Indeed, in this context, the
games have considerably different set of equilibria. As Shapley (1976, p.168) noted, any Nash
equilibrium of any buy or sell game is also a Nash equilibrium for the corresponding buy and sell game.
Conversely, a Nash equilibrium of the buy and sell game that happens to satisfy the buy or sell
restriction is obviously a Nash equilibrium of the buy or sell game. However, the hold-back option of
the buy and sell mechanism makes the set of solutions usually infinite (Shapley and Shubik, 1977; p.
964). 
Consider for example, any (non-zero) Nash equilibrium of a buy and sell market game with only4
two players. The corresponding strategy profile of the buy or sell game is however not an equilibrium
as the only Nash equilibrium of the buy or sell game with two players is no trade. This is because in the
buy or sell game, for any player, the opponent’s strategy is not an interior point and hence a player can
grab everything on the market by putting an infinitesimal amount. 
In fact, even when we impose strictly positive opponents’ bid (SPOB) condition, the set of
achievable allocations for any player in the buy or sell game, given others’ strategies, is not the same
as that in the original buy and sell game. Hence the games, buy and sell, buy or sell, are not individual
decision equivalent (Ray, 1999). This is precisely the reason why the equilibrium set of the buy and sell
game is considerably larger than that of the corresponding buy or sell game, although the games are
payoff equivalent. 
We construct an example (Example 2) to illustrate all the above remarks explicitly. This example
itself should be of interest as we do not have many clarifying examples in the market game literature.
2. THE GAME
In this section, we present an exchange economy as a non-cooperative game, namely, strategic
market game. We closely follow Dubey and Shubik (1978) in this section.
Consider an exchange economy with n agents and (l + 1) goods. Agents are indexed by i; i = 1,...,
n. First l goods are indexed by j; j = 1,..., l. The last good is money and is indexed by m. Endowment
of agent i is denoted by wi = (wi1,...,wil, wim) and the utility function of individual i is given by Ui : Ü
l
+ 1  Ü. Let us make standard assumptions that the endowments are strictly positive (wi >> 0 for all i)
and Ui is continuously differentiable, strictly increasing in all coordinates and strictly concave, for all
i.
To describe the strategic game, the agents in the economy are the players in the game. In a market
where the players are allowed to both buy and sell, the pure strategy set of player i is given by Si = {(qi,5
bi) : 0  qij  wij, 0  bij  for j = 1, ..., l and j=1
l bij  wim}. If the players only buy or sell goods, the pure
strategy set of player i is given by Si = {(qi, bi) : qij  . bij = 0, 0  qij  wij, 0  bij  for j = 1, ..., l and j=1
l
bij  wim}. Given a strategy profile, (q, b), prices are formed following the mechanism :
pj = total bid / total supply = i bij / i qij for j = 1, ..., l  ( = 0 if i qij = 0). Money is the numeraire.
The net allocations resulting from the strategy profile (q, b) are as follows. For j = 1, ..., l, 
xij (q, b) = wij - qij + bij / pj if pj > 0
= wij - qij if pj = 0
and,  xim (q, b) = wim - j=1
l bij + j=1
l pj qij.
To complete the game, the payoff of player i is ui (q, b) = Ui (xi(q, b)).
A non-cooperative equilibrium is a strategy profile, (q
*, b
*), such that for all i, for all (qi, bi)  Si,
ui (q
*, b
*)  ui ((q-i
*, b-i
*), (qi, bi)). This is nothing but a Nash equilibrium in pure strategies.
2. 1. The Set of Achievable Allocations
In this framework, one can easily characterise the set of achievable allocations for player i, given
the strategies of other players, (q-i, b-i). If player i plays a strategy (qi, bi) against (q-i, b-i), the price of
good j would be  pj = ((B-i)j  + bij) / ((Q-i)j + qij), where, (B-i)j = -i bij and (Q-i)j = -i qij. 
For a strictly positive price, the final allocation for player i would then be xij = wij - qij + bij / pj and
xim = wim - j=1
l bij + j=1
l pj qij. Observe that xij = wij + (Q-i)j - (Q-i)j - qij + bij / pj = wij + (Q-i)j - ((B-i)j  +
bij)/pj  + bij / pj = wij + (Q-i)j - (B-i)j /pj. Similarly, xim = wim + j=1
l (B-i)j - j=1
l (Q-i)j pj. Thus, given others’
strategies (q-i, b-i) and hence, (Q-i)j and (B-i)j, the final allocation of player i depends only on the price,
p. So, the final allocation remains same as long as p is constant. 
2. 2. Two Goods 
If there are only two goods, the strategy set of an individual i, having endowment wi = (wi1, wi2),
can be shown easily. Consider Figure 1. For buy and sell model, the pure strategy set of player i is     
1x = w1 and y = w2 satisfy the above equation implying that the hyperbola passes through the endowment point, (w1, w2). One
can also check that the hyperbola cuts the axes outside the rectangle OMWG as shown in Figure 1.
     
2Some of these observations and Figure 1 can also be found in other articles such as Dubey and Shubik (1978), Rogawski
and Shubik (1986). 
6
simply the rectangle OMWG constructed by the point W and the axes. For buy or sell model, the pure
strategy set of this player is given by the L-shaped part of the axes (MO U OG). 
The final allocation for any player, (x, y), is same when p = (B + b) / (Q + q) = constant, which
is an equation of a straight line passing through the point (-Q, -B). The set of achievable allocations
depends only on the total offers and bids made by the opponents, (Q, B). If both Q and B are strictly
positive, the set of achievable allocations, i.e, the locus of the point (x, y) is given by the quadratic
equation: xy - (w2 + B)x - (w1 + Q)y + (w1B + w2Q + w1w2) = 0, which is a hyperbola
1, as in Figure 1.
[ Insert Figure 1 here ]
Note that in a buy and sell game, any point (x, y) on this hyperbola can be obtained by a whole
straight line passing through the point (-Q, -B). Any such straight line must cut either of the axes and
this point on the axis generates  allocation (x, y) in a buy or sell game. 
If either Q or B is zero, then the set of achievable allocations is considerably different. For
example, if B is zero, then a player can achieve the endowment point (by offering and bidding nothing),
the good-axis (by offering something but bidding nothing and thereby losing the offer as the market
would not open) and a straight line, parallel to the money-axes (by any other strategy, grabbing all the
goods offered by others), characterised by x = w1 + Q and y = w2 - Qb/(Q + q). All the above
observations can be translated to the multi-good case.
2 
It is now easy to find the best response of player i against others’ strategies. The optimum
allocation would be the point of tangency between an indifference curve and the set of achievable
allocations. Thus for buy and sell game, the optimum allocation is unique however the best response7
is a set, the straight line that generates the optimum allocation.
3. RESULTS
The first question we are asking is whether or not an outcome of a buy and sell game can be
achieved as an outcome of a buy or sell game. Consider an outcome of a buy and sell game generated
by a strategy profile (q, b). Let us first observe the following. 
Observation 1. For any good j, the sum of the expression,  (B-i)j - pj(Q-i)j over all i is zero. 
Proof. i {(B-i)j - pj(Q-i)j} = (n-1)i bij  - pj  . (n-1)i qij = 0. 2
The above is not surprising at all. The price formation mechanism is such that the market clears
automatically. The net supply of good j for any player i is given by the amount ((B-i)j - pj(Q-i)j)/pj. Total
net supply for any good j is therefore zero. 
Given any strategy profile, (q, b), for any good j, if the expression (B-i)j - pj(Q-i)j is non-zero
(positive or negative) for one player, then there must exist another player for whom the expression is
also non-zero (negative or positive). Let us now consider the following condition.
Market-Opening (MO) : For every good j, the expression (B-i)j - pj(Q-i)j is non zero for at least one
player, where, pj = i bij / i qij, (B-i)j = -i bij and (Q-i)j = -i qij. 
For any strategy profile satisfying the above condition, by Observation 1, there must be at least
one player with a net supply of good j and one at least one with a net demand of good j. For  any such
strategy profile, the market for each good j opens in the corresponding buy or sell game; hence, we call
the above condition “market opening”.8
Proposition 1. Any outcome of a buy and sell game generated by a strategy profile (q, b) can be
obtained as an outcome of a buy or sell game if the profile satisfies MO.
Proof. Take any strategy profile (q, b) in a buy and sell game. Consider the following strategies for a
buy or sell game which generates the same outcome. Consider player i. For each good j, compute the
expression (B-i)j - pj(Q-i)j. If it is positive, then player i only sells the good j and qij1 = ((B-i)j - pj(Q-i)j)/pj.
If not, then individual i only buys good j with bij1 = pj(Q-i)j - (B-i)j. If it is zero, then the player neither
buys nor sells good j. 
If  the strategy profile (q, b) satisfies MO, then, for every good j, there exists at least one player
with qij1 > 0 and one with bij1 > 0 and hence, the market for good j opens. Note that in this buy or sell
game the price of good j is pj1 =   bij1 /  qij1 =  pj (by Observation 1). It is now obvious to check that
these strategies for buy or sell game generate the same allocation we have started with. 
If the strategy profile (q, b) does not satisfy MO, then, nobody would buy or sell some of the
goods and those market would not open. Still, the above mentioned strategies for buy or sell game
would generate the same allocation we have started with.     2
The above proposition proves that the two games, buy and sell, buy or sell are payoff equivalent
(Ray, 1999). The following example will illustrate the above the result. 
Example 1. Consider a market game with two players and two commodities only. Suppose the agents
have endowments W1 = (2, 5) and W2 = (5, 2) and any suitable utility function. 
Example 1a. Consider the strategies (q1 = 1, b1 = 2) and (q2 = 2, b2 = 1) in the buy and sell game. The
price of the good is 1. The expression B - Qp turns out to be -1 and 1 respectively. Therefore, the
strategy profile ((0, 1), (1, 0)) of the corresponding buy or sell game would generate the same outcome.9
Example 1b. Consider the strategies (q1 = 1, b1 = 1) and (q2 = 1, b2 = 1) for the buy and sell game. The
price is 1 in this case as well. This strategy profile does not satisfy MO and the outcome is generated
by the no-bid strategy, (0, 0), in the corresponding buy or sell game in which the market does not open.
We are now interested in any individual’s achievable outcomes. As we described in the previous
section, the set of achievable allocations of any player depends crucially on Q and B, the total bid and
offer made by the opponents. We therefore consider the following condition for a strategy profile (q,
b) of a buy and sell game.
Strictly Positive Opponents’ Bid (SPOB) : For every player i, for every good j, Q-ij1 = k ￿ i qkj1 and
B-ij1 = k ￿ i bkj1 are strictly positive, where qij1 and bij1 are the corresponding strategies in the buy or
sell game, as characterised in the proof of proposition 1. 
Remark 1. If a strategy profile in a buy and sell game does not satisfy MO, then it obviously does not
satisfy SPOB. For example, the strategy profile in Example 1.b. does not satisfy MO and thereby does
not satisfy SPOB. A profile satisfying MO, may also not satisfy SPOB. For example, the strategy profile
in Example 1.a does not satisfy SPOB. In fact, any strategy profile for any buy and sell game with 2 or
3 players would not satisfy SPOB.
Remark 2. Any (non-zero) strategy profile that does not satisfy SPOB would never be an equilibrium
of the buy or sell game as a player for which the condition is violated, can put an infinitesimal amount
to grab the all on the other side of the market.
We are now going to show that even when we impose SPOB, the equilibrium strategy profile of
any buy and sell game may not correspond to an equilibrium in the corresponding buy or sell game,
because the set of achievable allocations for any player may be different. The following example which10
uses a two-fold replica of the economy described in Example 1, clarifies all our remarks.
Example 2. Consider a market game with four players and two goods only. The first two agents are
identical in their endowments and preferences and the last two also have identical endowments and
preferences. Suppose the endowments are W1 = W2 = (2, 5), W3 = W4 = (5, 2) and the utility functions
are u1(x, y) = u2(x, y) = (15/16) ln x + ln y and u3(x, y) = u4(x, y) = (2/3) ln x + ln y.
Consider the strategies (q1 = 1, b1 = 2), (q2 = 1, b2 = 2), (q3 = 2, b3 = 1) and (q4 = 2, b4 = 1) in this
buy and sell game. It is easy to prove that this profile forms a Nash equilibrium. The strategy profile
indeed satisfies MO and SPOB. In the corresponding buy or sell game, the strategy profile ((0, 1), (0,
1), (1, 0), (1, 0)) generates the same outcome. However, it is not an equilibrium as either player can
deviate and be better off. The following table gathers all the characteristics of the example. 
Player 1 Player 2 Player 3 Player 4
Endowment : W 2, 5 2, 5 5, 2 5, 2
Utility : u(x, y) (15/16)ln x + ln
y
(15/16)ln x + ln
y
(2/3) ln x + ln y (2/3) ln x + ln y
Buy and Sell Game
Strategy : (q, b) 1, 2 1, 2 2, 1 2, 1
Allocation : (x, y) 3, 4 3, 4 4, 3 4, 3
Price = 1
Others’ : (Q, B) 5, 4 5, 4 4, 5 4, 5
Achievable set xy-9x-7y+43 = 0 xy-9x-7y+43 = 0 xy-7x-9y+43 = 0 xy-7x-9y+43=0
Best allocation 3, 4 3, 4 4, 3 4, 3
Buy or Sell Game
Strategy : (q1, b1) 0, 1 0, 1 1, 0 1, 0
Allocation : (x1, y1) 3, 4 3, 4 4, 3 4, 3
Price = 1
Others’ : (Q1, B1) 2, 1 2, 1 1, 2 1, 2     
3The best allocations for players 1 and 2 shown in the table are approximate values of the exact solutions which can be
obtained by solving the optimization problem : max (15/16)ln x + ln y subject to xy - 6x - 4y + 22 = 0.  
11
Achievable set xy-6x-4y+22 = 0 xy-6x-4y+22 = 0 xy-4x-6y+22 = 0 xy-4x-6y+22=0
Best allocation 2.8, 4.3 2.8, 4.3  
3 4, 3 4, 3
The above table clearly explains all the features of the relationship between the two games. Any
outcome, in particular, an equilibrium outcome, of the buy and sell game can be obtained as an outcome
of the buy or sell game; but the equilibrium outcome may not be an equilibrium in the buy or sell game,
because the individual’s set of achievable allocations may be different. 
The two sets of achievable allocations, ie, the two hyperbolas, must have (at least) two allocations
in common, the endowment point and the outcome in question. For example the hyperbolas xy-9x-
7y+43 = 0 and xy - 6x - 4y + 22 = 0 cut each other at points (2, 5) and (3, 4).
The equilibrium allocation is therefore still achievable by any player in the buy or sell game but
it may not be the best allocation for some of them. In the above example, for all four players, the two
sets of achievable allocations, ie, the two hyperbolas, are different; still, for players 3 and 4, the
allocation in question, (4, 3), is the best allocation they can achieve. However, for players 1 and 2,  the
allocation in question, (3, 4), is not the best allocation for them; they can, for example, play the strategy
(0, 0.7) to achieve a better allocation for them, given others’ strategies.
  12
References
1. Dubey, P., and Shubik, M.  1978 : "The Non-Cooperative Equilibria of a Closed Trading Economy
with Market Supply and Bidding Strategies." Journal of Economic Theory 17, 1-20.
2. Ray, I.   1999 : “On Games with Identical Equilibrium Payoffs.” Mimeo, University of York. 
3. Rogawski, J., and Shubik, M.  1986 : “A strategic Market Game with Transactions Costs.”
Mathematical Social Sciences 11, 139-160.
4. Shapley, L. S.  1976 : "Non-Cooperative General Exchange." In S. A. Y. Lin ed, Theory and
Measurement of Economic Externalities, pp. 155-175. Academic Press.
5. Shapley, L. S., and Shubik, M.  1977 : "Trade Using One Commodity as a Means of Payment."
Journal of Political Economy 85, 937-968.13

Non-Cooperative General Exchange.&quot; In

On Games with Identical Equilibrium Payoffs.” Mimeo,

The Non-Cooperative Equilibria of a Closed Trading Economy with Market Supply and Bidding Strategies.&quot;

Trade Using One Commodity as a Means of Payment.&quot;

Buying and Selling in Strategic Market Games

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Buying and Selling in Strategic Market Games

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