Social preferences, accountability, and wage bargaining

Abstract

We experimentally test preferences for employment in a collective wage bargaining situation with heterogeneous workers. We vary the size of the union and introduce a treatment mechanism transforming the voting game into an individual allocation task. Our results show that highly productive workers do not take employment of low productive workers into account when making wage proposals, regardless of whether only union members determine the wage or all workers. The level of pro-social preferences is small in the voting game, but it increases if the game becomes an individual allocation task. We interpret this as an accountability effect.

Appendices

Appendix A: Proofs of Propositions 1 and 2

Proof of Proposition 1

Suppose $0\le \uplambda <1$. To prove that the strategy profile $(\hbox {w}_{1},\hbox {w}_{2},\hbox {w}_{3})=(\hbox {Q}_{1},\hbox {Q}_{2},\hbox {Q}_{3})$ is a unique Nash equilibrium, we show that setting $\hbox {w}_{\mathrm{i}}=\hbox {Q}_{\mathrm{i}}$ is a strictly dominant strategy for worker i.

Suppose first worker 1 is decisive (i.e., worker 1’s wage has been randomly chosen). Then $\hbox {w}_{1}=\hbox {Q}_{1}$ is optimal if $100 >\hbox { max}\{50(2)^{\uplambda }, (100/3)(3)^{\uplambda }, 25(4)^{\uplambda },20(5)^{\uplambda }\}$. This holds when $\uplambda <1$. Similarly, when worker 2 is decisive, $\hbox {w}_{2}=\hbox {Q}_{2}$ is optimal when $50(2)^{\lambda }>\max \{(100/3)(3)^{\lambda },25(4)^{\lambda },20(5)^{\lambda }\}$. This again holds when $\uplambda <1$. Finally, when worker 3 is decisive, $\hbox {w}_{3}=\hbox {Q}_{3}$ is optimal when $(100/3)(3)^{\lambda }>\max \{25(4)^{\lambda },20(5)^{\lambda }\}$. Once more, this is satisfied when $\uplambda <1$.

We next show that under the median rule worker i receives a payoff from $\hbox {w}_{\mathrm{i}}=\hbox {Q}_{\mathrm{i}}$ that is at least as high as, or strictly higher than, the payoff from any other choice of $\hbox {w}_{\mathrm{i}}$, in any strategy profile. This follows from the fact that, whatever the other workers’ choices, worker i moves the median wage closest to her optimal value, which, when $0\le \uplambda <1$, is $\hbox {Q}_{\mathrm{i}}$, by setting $\hbox {w}_{\mathrm{i}}=\hbox {Q}_{\mathrm{i}}$ (other values of $\hbox {w}_{\mathrm{i}}$ may be equally optimal, but $\hbox {w}_{\mathrm{i}}=\hbox {Q}_{\mathrm{i}}$ is always among the set of optimal choices). To be more precise, suppose the two other workers set their wages below or equal to $\hbox {Q}_{\mathrm{i}}$. Then any wage $\hbox {w}_{\mathrm{i}}$ equal to or above the largest of these wages is optimal. Second, suppose one of the other workers sets his wage below $\hbox {Q}_{\mathrm{i}}$ and the other worker sets his wage above or equal to Qi. Then a unique best reply is $\hbox {w}_{\mathrm{i}}=\hbox {Q}_{\mathrm{i}}$. Finally, suppose each of the two other workers set their wage equal to or above $\mathrm{Q}_i$. Then any wage $\hbox {w}_{\mathrm{i}}$ is optimal. It follows that under the median rule $\hbox {w}_{\mathrm{i}}=\hbox {Q}_{\mathrm{i}}$ is a best reply, regardless of the wage choices of the two other workers. Combining this with the analysis for when the worker is decisive shows that setting $\hbox {w}_{\mathrm{i}}=\hbox {Q}_{\mathrm{i}}$ is a strictly dominant strategy, and hence the profile $(\hbox {w}_{1},\hbox {w}_{2},\hbox {w}_{3})=(\hbox {Q}_{1},\hbox {Q}_{2},\hbox {Q}_{3})$ is the unique Nash equilibrium.

Suppose now $\uplambda =1$. When worker i is decisive, each of the wages $\{\hbox {Q}_{\mathrm{i}},{\ldots },\hbox {Q}_{5}\}$ is a best reply. This implies that any profile $(\hbox {w}_{1},\hbox {w}_{2},\hbox {w}_{3})$ such that $\hbox {w}_{\mathrm{i}}\in \{\hbox {Q}_{\mathrm{i}},{\ldots },\hbox {Q}_{5}\}$ is a Nash equilibrium, because when worker i is decisive any wage $\hbox {w}_{\mathrm{i}}\in \{\hbox {Q}_{\mathrm{i}},{\ldots },\hbox {Q}_{5}\}$ is optimal. Under the median rule, any wage $\hbox {w}_{\mathrm{i}}\in \{\hbox {Q}_{\mathrm{i}},{\ldots },\hbox {Q}_{5}\}$ is once more optimal. To see this, suppose first that each of the two workers has chosen a wage below or equal to $\hbox {Q}_{\mathrm{i}}$. In this case any wage $\hbox {w}_{\mathrm{i}} \in \{\hbox {Q}_{1},{\ldots },\hbox {Q}_{5}\}$ is optimal for worker i. Suppose then one worker has chosen a wage below Qi, and the other has chosen a wage above or equal to $\mathrm{Q}_i$. Then any wage $\hbox {w}_{\mathrm{i}} \in \{\hbox {Q}_{\mathrm{i}},{\ldots },\hbox {Q}_{5}\}$ is optimal. Finally, suppose both other workers choose a wage equal to or above $\hbox {Q}_{\mathrm{i}}$. Then, any wage $\hbox {w}_{\mathrm{i}}\in \{\hbox {Q}_{1},{\ldots },\hbox {Q}_{5}\}$ is optimal for worker i. This implies that no matter what the other workers decide, any wage $\hbox {w}_{\mathrm{i}} \in \{\hbox {Q}_{\mathrm{i}},{\ldots },\hbox {Q}_{5}\}$ is optimal under the median rule. Any profile $(\hbox {w}_{1},\hbox {w}_{2},\hbox {w}_{3})$ such that $\hbox {w}_{\mathrm{i}}\in \{\hbox {Q}_{\mathrm{i}},{\ldots },\hbox {Q}_{5}\}$ is therefore a Nash equilibrium.

Suppose now that $\uplambda >1$. We show that setting $\hbox {w}_{\mathrm{i}}=\hbox {Q}_{5}$ is a strictly dominant strategy for each worker. When decisive, $\hbox {w}_{1}=\hbox {Q}_{5}$ is optimal for worker 1 when $20(5)^{\lambda }>\max \{100,50(2)^{\lambda },(100/3)(3)^{\lambda },25(4)^{\lambda }\}$. This holds when $\uplambda >1$. When worker 2 is decisive, $\hbox {w}_{2}=\hbox {Q}_{5}$ is optimal if $20(5)^{\lambda }>\max \{50(2)^{\lambda },(100/3)(3)^{\lambda },25(4)^{\lambda }\}$, and this again holds when $\uplambda >1$. Finally, consider worker 3. When he is decisive, $\hbox {w}_{3}=\hbox {Q}_{5}$ is optimal when $20(5)^{\lambda }>\max \{(100/3)(3)^{\lambda },25(4)^{\lambda }\}$, and this again holds when $\uplambda >1$.

Under the median rule, when $\uplambda >1$, worker i’s best outcome is that the median wage is $\hbox {Q}_{5}$. Thus, irrespective of the other workers’ wage choices, worker i will want to bring the median as close to $\hbox {Q}_{5}$ as possible; setting $\hbox {w}_{\mathrm{i}}=\hbox {Q}_{5}$ is always the, potentially non-unique, wage that accomplishes this objective. This shows that setting $\hbox {w}_{\mathrm{i}}=\hbox {Q}_{5}$ is a strictly dominant strategy for each worker i when $\lambda >1$. This, in turn implies that the Nash equilibrium $(\hbox {w}_{1},\hbox {w}_{2},\hbox {w}_{3})=(\hbox {Q}_{5}, \hbox {Q}_{5}, \hbox {Q}_{5})$ is unique when $\lambda >1$. $\square$

Proof of Proposition 2

Suppose $\uplambda <1$. The proof that the strategy profile $(\hbox {w}_{1},\hbox {w}_{2},\hbox {w}_{3,}\hbox {w}_{4,}\hbox {w}_{5})=(\hbox {Q}_{1},\hbox {Q}_{2},\hbox {Q}_{3,}\hbox {Q}_{4,}\hbox {Q}_{5})$ is a unique Nash equilibrium is a straightforward generalization of the proof of Proposition 1. Similarly, when $\uplambda =1$, the proof that there are multiple equilibria again follows from a generalization of the proof for the corresponding part of Proposition 1. Finally, suppose $\uplambda >1$. The proof that $(\hbox {w}_{1},\hbox {w}_{2},\hbox {w}_{3},\hbox {w}_{4},\hbox {w}_{5})=(\hbox {Q}_{5},\hbox {Q}_{5},\hbox {Q}_{5},\hbox {Q}_{5},\hbox {Q}_{5})$ is the unique equilibrium is once more a straightforward generalization of the three-worker analysis from Proposition 1. $\square$

Appendix B: Social preference elicitation task (based on Kerschbamer 2015)

Disadvantageous inequality block

LEFT		Your Choice (please mark)	RIGHT
You get	Passive agent gets		You get	Passive agent gets
15 Points	30 Points	LEFT $\bigcirc \bigcirc$ RIGHT	20 Points	20 Points
19 Points	30 Points	LEFT $\bigcirc \bigcirc$ RIGHT	20 Points	20 Points
20 Points	30 Points	LEFT $\bigcirc \bigcirc$ RIGHT	20 Points	20 Points
21 Points	30 Points	LEFT $\bigcirc \bigcirc$ RIGHT	20 Points	20 Points
25 Points	30 Points	LEFT $\bigcirc \bigcirc$ RIGHT	20 Points	20 Points

Advantageous inequality block

LEFT		Your Choice (please mark)	RIGHT
You get	Passive agent gets		You get	Passive agent gets
15 Points	10 Points	LEFT $\bigcirc \bigcirc$ RIGHT	20 Points	20 Points
19 Points	10 Points	LEFT $\bigcirc \bigcirc$ RIGHT	20 Points	20 Points
20 Points	10 Points	LEFT $\bigcirc \bigcirc$ RIGHT	20 Points	20 Points
21 Points	10 Points	LEFT $\bigcirc \bigcirc$ RIGHT	20 Points	20 Points
25 Points	10 Points	LEFT $\bigcirc \bigcirc$ RIGHT	20 Points	20 Points

1.1 Translation of choices in the distributional-preferences elicitation task into WTP

Disadvantageous Inequality Block (DIP)

In the DIB subject chooses LEFT for the first time in row	${{\varvec{WTP}}}^{d}$			proxy for ${\varvec{WTP}}^{d}$used
1	$+$0.5	$\le$ ${ WTP}^{d}$		$+$0.5
2	$+$0.1	$\le$ ${ WTP}^{d}$	$+$0.5	$+$0.3
3	$+$0.0	$\le$ ${ WTP}^{d}$	$+$0.1	$+$0.05
4	−0.1	$\le$ ${ WTP}^{d}$	−0.0	−0.05
5	−0.5	$\le$ ${ WTP}^{d}$	−0.1	−0.3
never		${ WTP}^{d}$	−0.5	−0.5

Advantageous inequality block (AIB)

In the AIB subject chooses LEFT for the first time in row	${{\varvec{WTP}}}^{a}$			proxy for ${{\varvec{WTP}}}^{a }$ used
1		${ WTP}^{a}$ $\le$	−0.5	−0.5
2	−0.5	${ WTP}^{a}$ $\le$	−0.1	−0.3
3	−0.1	${ WTP}^{a}$ $\le$	−0.0	−0.05
4	$+$0.0	${ WTP}^{a}$ $\le$	$+$0.1	$+$0.05
5	$+$0.1	${ WTP}^{a}$ $\le$	$+$0.5	$+$0.3
Never	$+$0.5	$<{ WTP}^{a}$		$+$0.5

${{\varvec{WTP}}}^{d}$ for ${ WPT}^{d} > 0: {\vert }{} { WTP}^{d}{\vert } =$ amount of own material payoff the decision maker is willing to give up in the domain of disadvantageous inequality in order to increase the other’s material payoff by one unit;

For ${ WPT}^{d}< 0: {\vert }{} { WTP}^{d}{\vert } =$ amount of own material payoff the decision maker is willing to give up in the domain of disadvantageous inequality in order to decrease the other’s material payoff by one unit (in this interpretation inequalities need to be reversed; for instance, subjects who never switch on the X-list reveal that they are willing to give up at least 50 Cents of their own income to decrease the income of the other player by 1 Euro);

${{\varvec{WTP}}}^{a}$ defined similarly for the domain of advantageous inequality.