Institutions and information in multilateral bargaining experiments

1 Equilibrium predictions for the alternating offer treatments

We find that the alternating-seller treatment has a unique sub-game perfect Nash equilibrium, but the alternating-buyer treatment does not. To illustrate, let S be the surplus available at the start of bargaining such that S = V − (c₁ + c₂). We assume offers and demands are symmetric. That is, we assume that when a buyer makes offers to the sellers the offers are equal, and that when sellers make demands, the demand made by each seller is equal. Equal seller demands require the assumptions necessary for a Nash bargaining solution (Pareto optimality, independence of irrelevant alternatives, symmetry and invariance to linear transformations of utility as in Nash [1950]).

In the alternating-seller treatment because the game has an even number of periods, sellers propose in the first period and the buyer proposes in the last period, so the equilibria assume an even number of bargaining periods. Starting in the final period (period 10 in our experiments), backwards induction leads to the sub-game perfect Nash equilibrium where – in period 1 – the sellers makes symmetric demands () as described below in eq. [3] and the buyer accepts both.

With the introduction of costly delay, when the sellers make offers (in odd numbered periods), the sellers offer the buyer more than he would make by holding out until the final period. This makes it possible for the sellers to capture some of the surplus. To illustrate, consider a simple alternating-seller two-period, two-seller example with the same values and costs used in the experiments. In this case, the discount rates to determine real payoffs are in period 1 and in period 2 (see eq. [4]). In period 2, the buyer makes offers and the sellers accept or reject these offers. Since it is the final period, the buyer simply offers each zero (or slightly more) and the sellers both accept since slightly more than zero is better than zero. The buyer’s real payoff – due to discounting – is . In period 1, the sellers make demands of the buyer with the knowledge that if the buyer rejects both, he can earn $15 in period 2.

They also know that the buyer has four strategy choices – accept both demands (A,A), reject both demands (R,R), or accept one and reject one (A,R) and (R,A) – as illustrated in Table 3. Given the parameters of our two-period example and the SPE solution outlined in eq. [3], the sellers should each demand $6 of the surplus. If the buyer accepts both $6 demands, then each seller would earn $6 in real terms and the buyer would earn $30−$6−$6 = $18 because . This is clearly better from the buyer’s perspective than rejecting both and earning $15. However, what if the buyer accepts one seller’s demand (say seller 1) and rejects the other? Assuming the sellers each make demands of $6, if the buyer rejects one while accepting the other, then the game moves to period 2 and the buyer can and will demand the remaining surplus ($30−$6 = $24). The discount factor applied to all payoffs is , since real payoffs are determined by the average number of periods it takes to reach a complete set of agreements. Thus, the real payoffs for this situation would be: buyer = $24(.75) = $18, seller 1 = $6(.75) = $4.50, seller 2 = $0. Therefore, the dominant strategy (shown in Table 3 in bold) is to accept both.¹³ In fact, the equilibrium seller demand shown in eq. [3] follows from , where the LHS is the discounted payoff to the buyer from accepting all the seller demands and the RHS is the discounted payoff from rejecting one while accepting the others, thus ensuring that the seller demands are just high enough to make (A,A) the dominant strategy. To illustrate, suppose both sellers demand more, say $7. Repeating the same calculations as above results in the real payoffs shown in column 3 of Table 3, clearly, the buyer’s dominant strategy is to accept one and reject the other given the higher payoff. Note that this also results in lower payoffs for both sellers relative to the equilibrium solution in column 2. The next two columns illustrate that there is no incentive for sellers to deviate from the symmetric equilibrium demands by either offering more or less. Offering more results in a zero payoff for the deviating seller, while offering less results in a lower payoff for the deviating seller and does not impact the buyer’s decision to accept both. As illustrated, is the minimum seller demand required to get the buyer to accept both demands. When sellers make demands (odd periods):

They demand

[3]

where

[4]

[5]

The simple two-period, two-seller example described above can illustrate the odd period play by the sellers and thus the equilibrium solution for period 1, but to illustrate even period play by the sellers and the interaction between it and odd period play, we expand the above example to four periods. We maintain the same values and costs, but in this case, the discount rates to determine real payoffs are , , and for periods 1 through 4, respectively (see eq. [3]). As before, in the final period, the buyer simply offers each seller zero (or slightly more) and the sellers both accept. The buyer’s real payoff – due to discounting – is . In period 3 then, the sellers make demands of the buyer with the knowledge that if the buyer rejects both, he can earn $7.50 in period 4. As before, they know that the buyer has four strategy choices – accept both demands (A,A), reject both demands (R,R), or accept one and reject one (A,R) and (R,A) – and given the parameters of our example the SPE solution outlined in eq. [3] is for the sellers to each demand $6 of the surplus. See Table 4 for the real payoffs.

If the buyer accepts both $6 demands, then each seller would earn $3 in real terms and the buyer would earn $9 because . As before, this is clearly better from the buyer’s perspective than rejecting both and earning $7.50. However, what if the buyer accepts one sellers demand (say seller 1) and rejects the other? Assuming the sellers each make demands of $6, if the buyer rejects one while accepting the other, then the game moves to period 4 and the buyer can and will demand the remaining surplus ($30−$6 = $24). The discount factor applied to all payoffs is since real payoffs are determined by the average number of periods it takes to reach a complete set of agreements. Thus, the real payoffs for this situation would be: buyer = $24(.375) = $9, seller 1 = $6(.375) = $2.25, seller 2 = $0. Therefore, as before, the (weakly) dominant strategy (in bold) is to accept both.

For period 2, the buyer makes offers to the sellers with the knowledge that sellers can earn $3 each in discounted terms in the following period. Suppose that the buyer offers each seller $4 and the sellers both accept, then each would earn $3 in real terms because . Each seller would seem to have an incentive to accept this offer since they would earn the same as in the following period, but in fact each seller has a dominant strategy to reject as illustrated in the seller normal-form game in Figure 1.

Figure 1

Period 2 seller normal-form game when buyer offers each seller $4.

Assuming the buyer makes each seller an offer of $4, if one rejects (say seller1) while the other accepts, then the game becomes a bargaining game between the buyer and remaining seller. Backward inducting, in period 4 the buyer will demand the remaining surplus ($30−$4 = $26) and the seller will accept. The discount factor applied in this case is and the real payoff to the buyer would be $9.75. In period 3, the remaining seller would offer the buyer an amount x such that the buyer would earn $9.75 after discounting – . Solving for x yields x = $15.60. Seller 1 would then take the remaining surplus ($26−$15.60 = $10.40) which when discounted results in a real payoff of $6.50, while seller 2, who accepted an offer of $4 would earn $2.50 after discounting. As noted above, each seller has a dominant strategy to reject, and the buyer ends up earning $9.

In order for the buyer to earn a greater payoff by getting the sellers to accept, he must offer the sellers more than the $3 (real payoff) they can earn in period 3. As the buyer increases the offers to the sellers, three changes occur to the seller payoffs in Figure 1: 1) the payoffs when both accept rise, 2) the payoff to the rejecting seller (when the other accepts) decreases – bolded payoffs in Figure 1 – because there is less surplus left in period 3, and 3) the payoff to the accepting seller increases – italicized payoffs in Figure 1 – due to being offered more. For example, if the buyer offers each seller $4.25 (instead of $4), then using the methods outlined above, the real payoffs would be as shown in Figure 2.

Figure 2

Period 2 seller normal-form game when buyer offers each seller $4.25.

Clearly, reject still dominates accept, but the difference in payoffs have shrunk from $3.50 ($6.50−$3) to $.03 ($3.22−$3.19) given the other seller accepts and from $.50 ($3-$2.50) to $.34 ($3−$2.66) when the other seller rejects. As one can see, the gap between payoffs when the other seller accepts is closing faster than the gap when the other seller rejects. It turns out that the “reject gap” is the limiting constraint to making accept the dominant strategy. The solution outlined in eq. [6] solves the buyer’s problem of choosing the lowest offer that results in changing the dominant strategy for sellers from reject to accept.

[6]

where:

Eq. [6] simply sets the real discounted payoff a seller gets if it accepts while the other(s) rejects (LHS) equal to the real discounted payoff if both (all) reject (RHS). Plugging the example parameters into eq. [6] yields . If both sellers accept, each earns a real payoff of $3.60 (), and the buyer earns a real payoff of $15.30 () (see Figure 3 below). If both reject, the real payoffs remain the same as before. If one rejects (say seller 1) and the other accepts, then the game becomes a one-to-one bargaining game and using the backward induction technique outline above, the real payoffs would be: seller 1 = $3.15, seller 2 = $3, buyer = $12.60. Clearly, the weakly dominant strategy for sellers is to accept.

Figure 3

Period 2 seller normal-form game when buyer offers each seller

However, as the reader may have noted, there exists a set of buyer offers where this normal-form game is a coordination game, that is, where neither player has a dominant strategy. This is because, as the offer rises, the positive payoff difference from rejecting when the other seller accepts closes (and becomes negative) faster than the positive payoff difference from rejecting when the other seller rejects. For example, suppose the buyer offers each seller $4.50 – less than the $4.80 offer that causes accept to be the dominant strategy and more than the $4.25 offers where reject is the dominant strategy – then using the methods outlined above, the real payoffs would be shown in Figure 4.

Figure 4

Period 2 seller normal-form game when buyer offers each seller $4.50.

Clearly, this is a coordination game – if seller 2 accepts, seller 1 should accept but if seller 2 rejects, seller 1 should reject. Eq. [7] solves for the offer above which the game becomes a coordination game.

[7]

where

is the buyer offer in period t.

Eq. [7] is derived from the idea that the buyer would have to make offers such that if both sellers accepted, they would earn the same discounted payoff they would get if they rejected, while other seller(s) did not and then bargained one-on-one with the buyer in the remaining periods. Plugging in for our parameters

and given, , and yields: .

If the buyer offers each seller $4.2857, then using the methods outlined above to solve for the real payoffs one can construct the normal-form game payoff matrix given in Figure 5. With this offer, reject is weakly dominant, but a slightly higher offer will make accept the dominant strategy given the other seller accepts via increasing the (accept, accept) payoff and lowering the (accept, reject) payoff.

Figure 5

Period 2 seller normal-form game when buyer offers each seller $4.2857 (minimum offer to create a coordination game).

Overall, as illustrated in Figure 6, one can conclude that when the buyer is making offers to the sellers in the alternating-seller institution (i.e. in even periods), the buyer will make offers greater than (see eq. [7]) and less than or equal to (see eq. [6]), unless it is the final period, in which case the buyer demands the entire surplus. This means that sellers will either face a coordination game or one where (Accept, Accept) is weakly dominant. This might appear to present an issue with regard to solving for a unique SPNE, but as explained below it does not.

Figure 6

Buyer bid ranges in even periods of alternating-seller treatments (bid increases to right).

Continuing with our four-period example, this means that in period 2 the buyer will make the sellers offers such that and the buyer would earn a real payoff between $9 (both sellers reject) and $16.07 (both sellers accept when offered $4.2857). For our purposes, the critical question is: Does this impact what sellers do (demand) in period 1? Turns out, no it does not. To illustrate this, solve for demand d₁ as outlined in the SPE solution in eq. [3].

Note that this is the same undiscounted demand we calculated for period 3 – in fact, regardless of number of periods, the sellers will always make the same demand – however, because of discounting, in real terms this is a higher demand. Recall that the buyer has four strategy choices – accept both demands (A,A), reject both demands (R,R), or accept one and reject one (A,R), and (R,A). See the Table 5 below for the real payoffs.

If the buyer accepts both $6 demands, then each seller would earn $6 in real terms and the buyer would earn $18 because . This is clearly better from the buyer’s perspective than rejecting both and earning somewhere between $9 and $16.07. What if the buyer accepts one sellers demand (say seller 1) and rejects the other? Using the backward induction technique discussed earlier, the real payoffs for this situation would be; buyer = $18, seller 1 = $5.25, seller 2 = $3. Therefore, as before, the (weakly) dominant strategy (in bold) is to accept both. The important aspect to note here is that the real payoff the buyer would earn in the next (or even) period – given he is bargaining with more than one seller – does not impact equilibrium seller demands because the relevant (or limiting) factor of consideration is how much the buyer can earn if he plays (Accept, Reject)¹⁴ and goes on to bargain with the rejected seller in subsequent periods. In other words, the seller demands that solve the problem of setting the buyer’s payoff if he plays (Accept, Accept) equal to his payoff if he plays (Accept, Reject), always result in buyer payoffs that are higher than those if he plays (Reject, Reject). As such, if the sellers make offers as shown in eq. [3] – or slightly higher, the buyer will accept both and the SPNE equilibrium for the alternating-seller game (for any even number of periods and any number of sellers greater than 2) is unique and expressed in eq. [3].

In the alternating-seller game, the buyer goes last and therefore enjoys an advantage if he decides to reject one seller’s offer while accepting the others. This is because the buyer is then bargaining one-on-one with the remaining seller and since he goes last, he can extract the majority of the remaining surplus. That is, the buyer enjoys the final-mover advantage and has the ability to take advantage of it and the sellers know this and subsequently have to offer him more to avoid that problem. This is why the relevant or limiting factor of consideration as noted above is how much the buyer would make if he accepts one seller’s demand and goes on to bargain with the other seller in subsequent periods. Put simply, this final-mover advantage and the ability to take advantage of it is why how much the buyer will make in the next period is irrelevant and can be ignored when determining the game-theoretic equilibrium. This is not the case in the alternating-buyer game. In that case, the sellers go last – meaning that if the buyer chooses to reject one seller’s offer while accepting the other’s the buyer no longer has the final-mover advantage. Thus, even though the buyer is bargaining one-on-one with the remaining seller, he can no longer extract the majority of the remaining surplus. As such, the sellers do not have to offer the buyer more to avoid this possibility and this possibility (or rather how much the buyer would make if he accepts one seller’s demand and goes on to bargain with the other seller in subsequent periods) is no longer the relevant or limiting factor of consideration. Instead the relevant factor of consideration for sellers as they make their demands in the alternating-buyer game is the amount that the buyer can make in the following period given he rejects both sellers’ demands. Unfortunately, whereas the amount the buyer can make if he rejects one and accepts the other and bargains one-on-one is easily calculated and (given a set of parameters) unique, the amount the buyer can make in the following period if he rejects both is not unique and is, in fact, dependent on beliefs about unknowable future buyer offers.

To illustrate this difference between the alternating-seller and alternating-buyer games, consider an alternating-buyer four-period, two-seller example with the same values and costs used in the example above and the experiments. In the final period, the sellers make demands and each ask for half of the surplus ($15 since S = $30) and the buyer accepts both. Each seller’s real payoff – due to discounting – is . Note that just as in the alternating-seller game, if in period N−1 (Period 3 here) the buyer makes offers to each seller such that they earn the same discounted payoff as they would earn in the final period ($3.75 here), each seller has a dominate strategy to reject (see Figure 7 – Note: if this was not the second to last (T−1) period, buyer would earn non-zero real payoffs due to costly delay).

Figure 7

Period 3 seller normal-form game when buyer offers each seller $3.75.

Since (Reject, Reject) is the dominant strategy, the buyer has to make higher offers if he wants to capture more of the surplus than he would earn in the next period ($0 here since both sellers reject). As in the alternating-seller game, there exists (as illustrated in Figure 8) a and where buyer offers between and result in sellers facing a coordination game regarding acceptance or rejection and buyer offers greater than or equal to result in (Accept, Accept) being the dominant strategy.

Figure 8

Buyer bid ranges in odd periods of alternating-buyer treatments.

Eq. [8] shows the solution for and is derived from the idea that the buyer must offer each seller enough of the surplus to make acceptance by sellers a weakly dominant strategy.

[8]

where is the minimum buyer offer in period t that makes (Accept, Accept) weakly dominant, S is the total surplus available to be split, N is the number of sellers (always one buyer)

[9]

[10]

Eq. [8] follows from , where the LHS is the discounted payoff from accepting while others accept and the RHS is the discounted payoff from rejecting while others accept.¹⁵

Eq. [10] shows that can be interpreted as the average potential discount factor given that N−1 sellers accept in period t (in our experiments where N = 2 this implies that one seller accepts, while the other rejects). The first term in the numerator represents the discount rates from the seller(s) who accept in period t, while the second term represents the discount rate for the seller who rejects in period t and is the average potential discount rate for the remaining periods. Summing these two terms and dividing by the number of sellers gives the average potential discount factor.

Eqs. [11] and [12] show the solution for – i.e. the minimum buyer offer to make the seller normal-form game a coordination game. Recall that is only well-defined/unique in the second to last (t = T−1) period.

[11]

where is the symmetric sellers’ demands in period t and is defined as

[12]

Note that we have defined seller equilibrium play in even periods in eq. [12]. Eq. [12] describes the idea that sellers can simply make demands such that the buyer is left with the same discounted payoff he would get in the following (t + 1) period. In period T eq. [12] becomes simply since it is the final period. It is important to note that these seller demands are fundamentally different than those described in the alternating-seller game where seller demands were NOT a function of buyer offers in subsequent periods. This is because here in the alternating-buyer game, sellers don’t need to worry about the possibility of the buyer rejecting one seller’s offer while accepting the others because the buyer does not enjoy the final mover advantage in thealternating-buyer game. Thus the sellers in thealternating-buyer game don’t need to offer the buyer more than he would earn in the next period whereas the buyers in thealternating-seller game must.

As one can see from eqs. [8], [11] and [12], while is well defined for all periods, is not since it is a function of seller demands in the next period, which are in turn a function of the buyer offer in the period after that, which are in turn a function of seller demands in the period after that, and so on. This would not be an issue (one could backward induct from the final period) except for the fact that the buyer offers in odd periods do not have a unique equilibrium solution which results in a quickly expanding strategy space as one backward inducts from period to period. Note that in periods in which the buyer is making offers (odd periods), the buyer is in essence making a decision involving trading off own-payoff with probability of seller acceptances. The buyer can ensure (assuming rational game-theoretic sellers) acceptance by making offers equal to , or he can decide to go for a higher payoff, ut at a lower probability of seller acceptances. This type of choice is based on the buyer’s risk preferences and his beliefs about what type of players the sellers are. Since these choices are based on unobservable beliefs we cannot offer a specific equilibrium prediction for the alternating-buyer game, but we can state that in the first period, buyers should never offer more than of the surplus to each seller, but may offer less. Using eq. [8], or is approximately 43% of the surplus. Thus, the maximum offer made to each seller should be $12.86 + $30 (cost of providing the unit) = $42.86. Since V = $90, the buyer would earn in this case $90 – $42.86 – $42.86 = $4.28 or ~14% of the surplus. If the buyer wants to ensure acceptance, he must offer the sellers ~ 86% of the surplus – clearly there is certainly room for the buyer to make lower offers that wouldn’t appear overtly greedy.