Nash equilibria with identical supports are compared for bimatrix games that are different with respect to the risk aversion of player 2. For equilibria in 2 by 2-bimatrix games and for equilibria with efficient supports in coordination games it is established for which cases increased risk aversion of player 2 benefits or hurts player 2.microeconomics ;

Berden,Caroline

Peters,Hans

English

Research Papers in Economics

On the e®ect of risk aversion in bimatrix games
Caroline Berden¤ Hans Peters¤
August 2005
Abstract
Nash equilibria with identical supports are compared for bimatrix
games that are di®erent with respect to the risk aversion of player
2. For equilibria in 2 £ 2-bimatrix games and for equilibria with e±-
cient supports in coordination games it is established for which cases
increased risk aversion of player 2 bene¯ts or hurts player 2.
1 Introduction
Uncertainty plays a central role in the theory of games, be it uncertainty
about the parameters of the game|called incomplete information|or uncer-
tainty about the actions of the opponent(s)|called strategic uncertainty by
von Neumann and Morgenstern (1944). Following these authors, uncertainty
is usually modelled by assuming the players to maximize expected utility.
It is somewhat of a surprise that the e®ects of one of the most extensively
studied characteristics of expected utility, namely the Arrow-Pratt measure
of risk aversion (Arrow, 1971; Pratt, 1964), have received relatively little
attention in game theory, with the exception of bargaining theory (starting
with Kannai, 1977, and Kihlstrom et al., 1981).
In the present paper we study the e®ects of increased risk aversion in
two-person noncooperative games with ¯nite pure strategy sets, i.e., bima-
trix games. Speci¯cally, we consider the situation where player 2 is replaced
¤Department of Quantitative Economics, University of Maastricht, P.O. Box 616, 6200
MD Maastricht, The Netherlands. Tel.: +31 43 3883835, email: c.berden@ke.unimaas.nl,
h.peters@ke.unimaas.nl.
1by a more risk averse player and study the e®ect of this on Nash equilib-
rium. In doing so we assume that the supports of the Nash equilibria under
comparison in the two situations|one with the less risk averse player 2 and
the other one with the more risk averse player 2|do not change, in order to
make a meaningful comparison. This implies that the equilibrium strategy of
player 2 and therefore the expected payo® of player 1 does not change. This
may appear peculiar but it is a direct consequence of the fact that in a mixed
strategy Nash equilibrium a player is indi®erent between the pure strategies
that are played with positive probability. Since it is meaningless to com-
pare the expected payo®s of player 2 in the two situations, we compare the
equilibria themselves. Speci¯cally, we say that risk aversion bene¯ts player
2 if the Nash equilibrium in the situation with the more risk averse player 2
would give the less risk averse player 2 a higher payo®. In the opposite case,
we say that risk aversion hurts player 2.
We give a complete analysis of the 2£2-case (Section 3), and of coordina-
tion games in which the players coordinate on Nash equilibria with e±cient
supports (Section 4). The analysis of the 2£2-case suggests that it is di±cult
to obtain results for the completely general case.
We identify cases where risk aversion bene¯ts player 2|this happens in
particular for Nash equilibria with e±cient supports in coordination games|
and cases where risk aversion hurts player 2. In the former case, as an
alternative interpretation, it would be advantageous for player 2 if he could
make player 1 believe that he is more risk averse than he actually is. In the
latter case, in contrast, player 2 would want player 1 to believe that he is
less risk averse than he actually is. See also the discussion in Section 2.
After preliminaries in Section 2 we proceed with 2 £ 2 games in Section
3 and coordination games in Section 4. Section 5 concludes.
2 Preliminaries
A bimatrix game is a pair (A;B) of m£n-matrices of real numbers. A (mixed)
strategy for player 1 is an element p 2 ¢m := fx 2 Rm j x ¸ 0;
Pm
i=1 xi = 1g.
Similarly, a strategy for player 2 is an element q 2 ¢n. The strategy pro¯le
(p;q) results in the expected payo®s pAq for player 1 and pBq for player 2.
A Nash equilibrium is a strategy pro¯le (p¤;q¤) such that p¤Aq¤ ¸ pAq¤ for
all p 2 ¢m and p¤Bq¤ ¸ p¤Bq for all q 2 ¢n. Let aij (bij) denote the element
in row i and column j of matrix A (B). As is well known, (p¤;q¤) is a Nash
2equilibrium if and only if
n X
j=1
aijq
¤
j ¸
n X
j=1
akjq
¤
j for all i with p¤
i > 0 and all k = 1;:::;m (1)
and
m X
i=1
bijp
¤
i ¸
m X
i=1
bikp
¤
i for all j with q¤
j > 0 and all k = 1;:::;n: (2)
Let (A;B) be a bimatrix game and let k be a concave and strictly increasing
function de¯ned on (at least) the interval [minfbijg;maxfbijg]. Let k(B)
denote the matrix with element k(bij) in row i and column j. We say that
player 2 in the game (A;k(B)) is more risk averse than player 2 in the game
(A;B) (cf. Pratt, 1964).
Suppose, in this situation, that (p¤;q¤) is a Nash equilibrium in (A;B) and
that (~ p;q¤) is a Nash equilibrium in (A;k(B)) such that p¤ and ~ p have the
same support, that is, p¤
i > 0 , ~ pi > 0 for all i = 1;:::;m. We say that risk
aversion bene¯ts player 2 at (p¤;q¤) and (~ p;q¤) if
~ pBq
¤ ¸ p
¤Bq
¤ (3)
and that risk aversion hurts player 2 at (p¤;q¤) and (~ p;q¤) if
~ pBq
¤ · p
¤Bq
¤: (4)
In words, if risk aversion bene¯ts [hurts] player 2, then the equilibrium in
the game with the more risk averse player 2 is better [worse] for the less risk
averse player 2 than the equilibrium in his own game.
Note that we assume that the two Nash equilibria under comparison are
similar in the sense that their supports are the same. Hence, the players
agree on the strategies that should be played with positive probability. This
implies that player 2 should not (or, at least, must not) change his strategy
(q¤ in both equilibria), so that (1) is ful¯lled in both (A;B) and (A;k(B)).
Player 1, however, adapts his strategy from p¤ in (A;B) to ~ p in (A;k(B)), in
order for (2) to hold with ~ p and k(B) instead of p¤ and B.
Also note that it does not make sense to compare the expected payo®s
p¤Bq¤ and ~ p k(B)q¤, since the numbers in B and k(B) represent the under-
lying preferences only up to a positive a±ne transformation.
3An alternative justi¯cation for de¯nitions (3) and (4) is that player 1
may have only limited information about the utility function of player 2.
Speci¯cally, suppose that player 1 knows player 2's utility function up to an
increasing concave transformation, i.e., up to player 2's risk attitude. To
¯x ideas, one might think of the matrices B and k(B) as representing the
utilities of monetary outcomes. Player 1 knows these monetary outcomes
but not the utilities attached to them by player 2. The inequalities in (3)
and (4) re°ect the possible consequences for player 2 in a Nash equilibrium if
player 1 believes that player 2 is more risk averse than he actually is. For this
interpretation we must implicitly assume that player 2 behaves myopically in
the sense that he does not react optimally to the `wrong' strategy of player
1, that is, he does not play a strategy ~ q that maximizes his payo® ~ pBq over
all q 2 ¢n. Allowing for this possibility would only increase the e®ect in
(3), but might undo the e®ect in (4). This assumption of myopic behavior
makes sense if player 2 does not know that player 1 misjudges player 2's risk
attitude. But even if player 2 tries to make player 1 believe that he (player
2) is more risk averse than he actually is|because risk aversion may bene¯t
player 2|then he might still behave myopically in order not to reveal his
true risk attitude, although making this argument precise would call for a
dynamic model such as a repeated game.
The purpose of this note is to investigate in which cases risk aversion
bene¯ts or hurts player 2. In the next section we analyze the 2 £ 2-case,
and in Section 4 we consider the special case of coordination games. As the
2 £ 2-case will reveal, a complete analysis of the general case does not seem
to be tractable.
3 2 £ 2-bimatrix games
We consider bimatrix games (A;B) with m = n = 2. Let k be a strictly
increasing concave transformation. For pure strategy Nash equilibria (p¤;q¤)
in (A;B) we have, in the notation of the preceding section, ~ p = p¤, so that
(3) and (4) are ful¯lled with equality. This case is not interesting and, in
fact, we will assume,a forteriori, that player 2 has no (weakly) dominated
strategy. For
B =
·
b11 b12
b21 b22
¸
4this implies that b11 6= b12. Without loss of generality we take b11 > b12
and, thus, b21 < b22. It is easy to verify that the following|not completely
disjoint|cases are exhaustive:
(i) b11 > b12 ¸ b22 > b21;
(ii) b11 ¸ b22 ¸ b12 ¸ b21 and b11 > b12, b22 > b21;
(iii) b11 ¸ b22 > b21 ¸ b12.
Let (p¤;q¤) be a completely mixed Nash equilibrium in (A;B), that is 0 <
p¤
1;q¤
1 < 1. (Of course, the existence of such an equilibrium depends on A as
well, but will just be assumed here.) Then
p
¤
1 =
b22 ¡ b21
b22 ¡ b21 + b11 ¡ b12
; (5)
as can easily be veri¯ed.
We ¯rst consider case (i), and normalize the function k such that k(b21) =
b21 and k(b11) = b11. Hence, k(b22) ¸ b22 and k(b12) ¸ b12 by concavity of k.
For the equilibrium (~ p;q) in (A;k(B)) we now derive
~ p1 =
k(b22) ¡ b21
k(b22) ¡ b21 + b11 ¡ k(b12)
¸ p
¤
1: (6)
Hence
~ pBq
¤ = ~ p1[q
¤
1b11 + q
¤
2b12] + ~ p2[q
¤
1b21 + q
¤
2b22]
¸ p
¤
1[q
¤
1b11 + q
¤
2b12] + p
¤
2[q
¤
1b21 + q
¤
2b22]
= p
¤Bq
¤; (7)
where the inequality follows from (6) and the fact that q¤
1b11+q¤
2b12 ¸ q¤
1b21+
q¤
2b22. We conclude that, in case (i), risk aversion bene¯ts player 2 at (p¤;q¤)
and (~ p;q¤).
Consider next case (ii). The analysis of case (i) goes through including
(6), but now there are two cases:
(a) q¤
1b11 + q¤
2b12 ¸ q¤
1b21 + q¤
2b22. In this case, risk aversion again bene¯ts
player 2, just like in case (i), see (7).
5(b) q¤
1b11+q¤
2b12 · q¤
1b21+q¤
2b22. In this case we have the reverse inequality
in (7), so that risk aversion hurts player 2.
So in case (ii), whether risk aversion bene¯ts or hurts player 2 depends on
q¤, hence on the matrix A. Since
q
¤
1 =
a22 ¡ a12
a22 ¡ a12 + a11 ¡ a21
risk aversion bene¯ts player 2 (case (a)) if
a22 ¡ a12
a22 ¡ a12 + a11 ¡ a21
¸
b22 ¡ b12
b22 ¡ b12 + b11 ¡ b21
(8)
and risk aversion hurts player 2 (case (b)) if
a22 ¡ a12
a22 ¡ a12 + a11 ¡ a21
·
b22 ¡ b12
b22 ¡ b12 + b11 ¡ b21
: (9)
Consider, ¯nally, case (iii). For this case we show by means of an example
that the e®ect of risk aversion is ambiguous and depends not only on the
matrices A and B but also on the function k. Consider the bimatrix games
(A;B) =
·
0;7 1;0
1;2 0;4
¸
; (A;B
0) =
·
0;5 1;0
1;2 0;4
¸
; (A;B
00) =
·
0;7 1;¡1
1;2 0;4
¸
:
Clearly, both B0 and B00 can be obtained by applying increasing concave
transformations to B. The (unique) Nash equilibrium in (A;B) is (p¤;q¤) =
((2
9; 7
9); (1
2; 1
2)), in (A;B0) it is (p0;q¤) = ((2
7; 5
7);(1
2; 1
2)), and in (A;B00) it is
(p00;q¤) = ((1
5; 4
5);(1
2; 1
2)). Then p¤Bq¤ = 31
9, p0Bq¤ = 31
7, and p00Bq¤ = 3 1
10.
Hence, risk aversion bene¯ts player 2 at (p¤;q¤) and (p0;q¤) but hurts player
2 at (p¤;q¤) and (p00;q¤).
The intuition for the di®erent results in these three cases is as follows. In
case (i), player 1 has to put more weight on the upper row in order to keep
the more risk averse player 2 indi®erent, but this row is unambiguously the
best row from the point of view of player 2, so risk aversion bene¯ts player 2.
In case (ii) player 1 again has to put more weight on the ¯rst row, but now
it depends on q¤, and thus on the matrix A, whether the ¯rst or second row
is better from the point of view of player 2. This results in the two cases (a)
and (b). In case (iii) it is not only ambiguous which one of the two rows is
6better from the point of view of player 2, but also the change in weight put
by player 1 on the ¯rst row is ambiguous, as is clear from the example.
We summarize and slightly extend these results in the following two
propositions. The ¯rst proposition summarizes cases (i) and (ii).
Proposition 1 Let m = n = 2, let (A;B) and (A;k(B)) be bimatrix games
with k an increasing and concave function, and let (p¤;q¤) and (~ p;q¤) be
completely mixed Nash equilibria in (A;B) and (A;k(B)), respectively.
(i) If b11 > b12 ¸ b22 > b21, then risk aversion bene¯ts player 2 at (p¤;q¤)
and (~ p;q¤).
(ii) If b11 ¸ b22 ¸ b12 ¸ b21 and b11 > b12, b22 > b21, then risk aversion
bene¯ts player 2 at (p¤;q¤) and (~ p;q¤) if (8) holds and risk aversion
hurts player 2 at (p¤;q¤) and (~ p;q¤) if (9) holds.
The second proposition considers case (iii).
Proposition 2 Let m = n = 2, let (A;B) be a bimatrix game, and let
(p¤;q¤) be a completely mixed Nash equilibrium. Let b11 ¸ b22 > b21 ¸ b12.
Then there are increasing concave transformations k and k0 and Nash equilib-
ria (~ p;q¤) in (A;k(B)) and (~ p0;q¤) in (A;k0(B)) such that risk aversion ben-
e¯ts player 2 at (p¤;q¤) and (~ p;q¤) but hurts player 2 at (p¤;q¤) and (~ p0;q¤).
Proof. Let k satisfy k(x) = x for all x · b22 and k(b11) = b22 + 1
2(b11 ¡ b22).
Then it is straightforward to verify (using (5)) that ~ p1 ¸ p¤
1. Let k0 satisfy
k0(x) = x for all x ¸ b21 and k0(b12) = b12 ¡ 1
2(b21 ¡ b12). Then it is easy
to verify (again using (5)) that ~ p0
1 · p¤
1. If (8) holds, then k and k0 are as
desired. If (9) holds, then the proof is complete by switching the roles of k
and k0. ¤
Proposition 2 in particular makes it clear that there is not much hope to
obtain general results for the m£n-case. In the next section we consider the
special but interesting case of coordination games.
4 Coordination games
A coordination game is an m£m-bimatrix game (A;B) such that aij = bij =
0 whenever i 6= j, and aii;bii > 0 for all i = 1;:::;m. Clearly, the players
7would want to coordinate on one or more diagonal elements. In such a game,
for each non-empty subset I µ f1;:::;mg there is a unique Nash equilibrium
with support I for each player, namely the strategy pro¯le (pI;qI) satisfying
pI
ibii = pI
jbjj and qI
iaii = qI
jajj for all i;j 2 I, pk = qk = 0 for all k = 2 I:
(10)
Observe that, for m = 2, such a coordination game falls under case (ii)
and under case (iii) in Section 3. In fact, it is a boundary case between
these two: Proposition 2 applies, but the function k0 de¯ned in its proof is
constant, so that it only depends on q¤ (i.e., on A) whether risk aversion
bene¯ts or hurts player 2.
In a coordination game equilibria of interest are those where ine±cient
diagonal outcomes are played with zero probability. More generally, we say
that a Nash equilibrium (pI;qI) has e±cient support if for all i;j 2 I, aii >
ajj implies bii < bjj.1 Clearly, if (pI;qI) has e±cient support and k is an
increasing concave function with k(0) = 0, then the equilibrium (~ pI;qI) in
(A;k(B)) also has e±cient support.2 Moreover, we have the following result.
Proposition 3 Let (A;B) be an m £ m-coordination game and let, for
some I µ f1;:::;mg, (pI;qI) be a Nash equilibrium with e±cient support in
(A;B). Let k be an increasing concave function with k(0) = 0 and let (~ pI;qI)
be a Nash equilibrium in (A;k(B)). Then risk aversion bene¯ts player 2 at
(pI;qI) and (~ pI;qI).
Proof. Without loss of generality suppose I = f1;:::;mg and 0 < b11 ·
::: · bmm. We write p instead of pI, ~ p instead of ~ pI, and q instead of qI.
Since (p;q) has e±cient support and by (10), we have 0 < q1 · ::: · qm,
so that
0 < q1b11 · ::: · qmbmm: (11)
By concavity of k, k(bii)=bii ¸ k(bjj)=bjj whenever i · j. By (10) this
implies
~ pi
~ pj
=
k(bjj)
k(bii)
·
bjj
bii
=
pi
pj
for all i;j 2 I with i · j: (12)
1Clearly, unless I is a singleton, such an equilibrium is neither ex ante nor ex post
e±cient. So the e±cient support requirement is a very weak e±ciency requirement.
2The condition k(0) = 0 is imposed in order that (A;k(B)) is formally a coordination
game. No generality is lost.
8For all `;j 2 I with ` > j, (12) implies
p1
p`
+ ¢¢¢ +
pj
p`
¸
~ p1
~ p`
+ ¢¢¢ +
~ pj
~ p`
hence
p`
p1 + ¢¢¢ + pj
·
~ p`
~ p1 + ¢¢¢ + ~ pj
hence
pj+1 + ¢¢¢ + pm
p1 + ¢¢¢ + pj
·
~ pj+1 + ¢¢¢ + ~ pm
~ p1 + ¢¢¢ + ~ pj
hence
1 ¡ (p1 + ¢¢¢ + pj)
p1 + ¢¢¢ + pj
·
1 ¡ (~ p1 + ¢¢¢ + ~ pj)
~ p1 + ¢¢¢ + ~ pj
which implies
p1 + ¢¢¢ + pj ¸ ~ p1 + ¢¢¢ + ~ pj for all j = 1;:::;m: (13)
De¯ne b00 := 0, then
m X
j=1
pjqjbjj =
m X
j=1
Ã
m X
i=j
pi
!
(qjbjj ¡ qj¡1bj¡1;j¡1)
·
m X
j=1
Ã
m X
i=j
~ pi
!
(qjbjj ¡ qj¡1bj¡1;j¡1)
=
m X
j=1
~ pjqjbjj; (14)
where the inequality follows from (11) and (13). Now (14) shows that risk
aversion bene¯ts player 2. ¤
As a note to this proof, observe that (13) states that ~ p ¯rst-degree stochas-
tically dominates p, and (14) is the familiar consequence that the expected
payo® under ~ p is higher than under p.
The intuition for Proposition 3 is that increased risk aversion of player
2 forces player 1 to put more weight on the outcomes that are better for
player 2. From the point of view of the original|less risk averse|player 2
the resulting distribution is preferable. This intuition (and result) is similar
to the result established for a bargaining context in KÄ obberling and Peters
(2003), although that paper assumes rank-dependent utility instead of ex-
pected utility.
95 Concluding remarks
The present paper has shown that at least in some interesting cases it is
possible to say something about the e®ect of the risk attitude of a player in a
noncooperative game. From a more general point of view the paper can also
be interpreted as studying the e®ect of one player having limited information
about the utility function of another player. We focus, however, on risk atti-
tude, which is one of the most important characteristics of an expected utility
function. Further research may concentrate on speci¯c economic games, such
as risky investments in portfolios.
References
1. Arrow, K.J. (1971): Essays in the Theory of Risk Bearing. Markham,
Chicago, Illinois.
2. Kannai, Y. (1977): \Concavi¯ability and constructions of concave util-
ity functions," Journal of Mathematical Economics, 4, 1{56.
3. Kihlstrom, R.E., A.E. Roth, and D. Schmeidler (1981): \Risk aversion
and solutions to Nash's bargaining problem," in: O. Moeschlin and D.
Pallaschke (eds.), Game Theory and Mathematical Economics. North
Holland, Amsterdam.
4. KÄ obberling, V., and H. Peters (2003): \The e®ect of decision weights
in bargaining problems," Journal of Economic Theory, 110, 154{175.
5. Pratt, J.W. (1964): \Risk aversion in the small and in the large,"
Econometrica, 32, 122{136.
6. von Neumann, J., and O. Morgenstern (1944): Theory of Games and
Economic Behavior. Princeton University Press, Princeton.
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On the effect of risk aversion in bimatrix games

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