A study of computational and human strategies in revelation games

Abstract

Many negotiations in the real world are characterized by incomplete information, and participants’ success depends on their ability to reveal information in a way that facilitates agreements without compromising their individual gain. This paper presents an agent-design that is able to negotiate proficiently with people in settings in which agents can choose to truthfully reveal their private information before engaging in multiple rounds of negotiation. Such settings are analogous to real-world situations in which people need to decide whether to disclose information such as when negotiating over health plans and business transactions. The agent combined a decision-theoretic approach with traditional machine-learning techniques to reason about the social factors that affect the players’ revelation decisions on people’s negotiation behavior. It was shown to outperform people as well as agents playing the equilibrium strategy of the game in empirical studies spanning hundreds of subjects. It was also more likely to reach agreement than people or agents playing equilibrium strategies. In addition, it had a positive effect on people’s play, allowing them to reach significantly better performance when compared to people’s play with other people. These results are shown to generalize for two different settings that varied how players depend on each other in the negotiation.

Appendix

Before proving Proposition 1, we will postulate the following lemma:

Lemma 1

Suppose that player $j$ uses response strategy $a_{j}^{2}(\omega ,t_j)$ (Eq. 2) to accept proposals $\omega$ at round 2. Then player $i$ is always indifferent between revealing and not revealing its type, regardless of the revelation decision of player $j$.

Proof

We will show that the the revelation decisions of player $i$ cannot affect its outcome in the game. The strategy $a_{j}^{2}(\omega ,t_j)$ of player $j$ to accept the proposal by $i$ at round 2 does not depend on $i$’s type. Hence the revelation decision of $i$ in round 0 does not affect its benefit from proposals in round 2. Suppose that player $i$ revealed its type. If it can get a more beneficial proposal from player $j$ by not revealing its type, then it will reject $j$’s offer and make the more beneficial proposal in round 2. A similar argument can be used to show that player $i$ can get its more preferred outcome when it does not reveal its type. Hence the revelation decision of $i$ does not affect its outcome.

We provide a proof to Proposition 1 by showing that the strategies specified in Sect. 4.1 correspond to a perfect Bayesian equilibrium given the sufficient conditions.

Proof

We denote the possible types for $j$ as $t_{j_1}$ and $t_{j_2}$ and the possible types for $i$ as $t_{i_1}$ and $t_{i_2}$. We prove that when one type of player $j$ is preferable to the other type (Eq. 6), there always exists a revelation equilibrium in which players reveal their goals for both types.¹⁰ In round 2, player $i$ makes the proposal $\omega _{i,{j}}^{2^{*}}(t_j)$ using Eq. 4. This proposal gives the maximum benefit for player $i$ among the proposals that are accepted by player $j$. Any proposal with higher benefit for $i$ will be rejected by player $j$.¹¹ Player $j$ also cannot benefit from deviation as rejecting the proposal incurs zero benefit to both players.

We proceed to round 1. We first specify why the belief update rules are part of a perfect Bayesian equilibrium. Suppose $j$ is of type $t_{j_2}$ and that this type is preferable for $j$ (without loss of generality). If $j$ revealed its type, then $i$ uses Bayes rule to assign probability 1 to type $t_{j_2}$, as specified in equilibrium. If $j$ does not reveal its type, then the resulting information set is off the equilibrium path. In this case $i$ makes an arbitrary belief update to assign probability 0 to type $t_{j_2}$. Using a similar argument, we can show that the belief update rules are part of a perfect Bayesian equilibrium when type $t_{j_1}$ is preferable to $j$. Suppose that player $i$ is of type $t_{i_1}$. If $i$ reveals its type, then $j$ uses Bayes rule to assign probability 1 to $t_{i_2}$, as specified in equilibrium. If $i$ does not reveal its type, then the resulting information set is off the equilibrium path. In this case $j$ makes an arbitrary belief update to assign probability 1 to $t_{i_1}$. Using a similar argument, we can show that the belief update rules are part of a perfect Bayesian equilibrium when $i$ is of type $t_{i_2}$.

Next, we specify why players’ actions are in equilibrium in round 1. Player $j$ makes the proposal $w^{1*}_{j,i}(t_i)$ using Eq. 5. Any proposal with a higher benefit to $j$ will necessarily be rejected by player $i$. As a result, player $j$’s proposal maximizes its benefit given $i$’s response strategy. Player $i$ also cannot benefit from deviation because player $j$’s proposal gives it the same benefit as its counter-proposal $\omega _{i,j}^{2*}(t_j)$.

Finally, we consider round 0, in which both players $i$ and $j$ reveal their types for both of their types. A deviation for player $j$ of type $t_{j_2}$ in round 0 means that $j$ chooses not to reveal its type. According to the belief-update rules, player $i$ will update its beliefs to assign probability 1 to type $t_{j_1}$. As a result, it will make a proposal in round 2 to $j$ that is less preferable to the one it would receive if it revealed its type. Thus, deviation from its revelation decision is not strictly beneficial for $j$ when it is of type $t_{j_2}$. In a similar fashion, we can show that deviation from a revelation decision is not beneficial for $j$ when it is of type $t_{j_1}$. Thus the revelation strategies of round 0 are in Equilibrium for $j$.

Due to Lemma 1, a decision by player $i$ to reveal its type in round 0 for each of its types is in equilibrium. Thus there always exists a revelation equilibrium in which players reveal their goals for both types.

We provide a proof to Proposition 1 by showing that the strategies specified in Sect. 4.2 specify a perfect Bayesian Equilibrium.

Proof

We use backward induction. In Round 2 there are two cases. On the equilibrium path. In this case the players did not update their beliefs according to the equilibrium specification.

Player $i$ makes the proposal $\omega _{i,{j}}^{2^{*}}(h^1,t_i)$ using Eq. 10. This proposal gives the maximum expected benefit for player $i$, so it cannot on expectation benefit from deviation. Player $j$ also cannot benefit from deviation as rejecting the proposal incurs zero benefit to both players.

Lets examine the cases where the information set is off the equilibrium path. The first case is related to round 0: if $i$ revealed its type, this will not affect the proposal at round 2 (Proposition 2). If $j$ revealed its type, then $i$ knows its true type, and the strategies are in equilibrium as in the revelation equilibrium. The other case is related to round 1: In this case, as defined in the Equilibrium, the players do not change their beliefs, and the strategies in round 2 are as in the equilibrium. We now proceed to Round 1. we show neither player has incentive to deviate from its strategy. Suppose player $j$ of type $t_j$ wishes to deviate and propose a different proposal. If the proposal is worse off for player $i$, given its type $t_i$, then player $i$ will reject the offer and player $j$ will not benefit. If player $i$ accepts the proposal (according to Eq. 12), this means that it is greater or equal in benefit to $i$ than the equilibrium proposal in round 2. By the second condition (Eq. 14), this means that the proposal will be at most equal to the benefit that player $j$ will receive from the equilibrium proposal. So player $j$ will not benefit from making this proposal. For player $i$, it cannot benefit from rejecting $j$’s proposal. According to Eq. 12, player $i$ accepts player $j$’s proposal only if it cannot get a higher benefit from its own counter-proposal.

Finally, Lets examine if the player can benefit from deviation in Round 0. Suppose that player $j$ of type $t_{j_1}$ deviates and reveals its type. By definition, then player $i$ will update its beliefs to its true type. In this case, player $i$ will make the proposal $\omega _{i,j}^{2^*}(t_{j_1})$. However, by the first condition (Eq. 13), this proposal is at most equal to the proposal that player $j$ will get from not revealing its type. Therefore it is not beneficial for player $j$ to deviate. By Proposition 1, player $i$ is indifferent between its two types. Thus the strategy for both players satisfies a perfect Bayesian equilibrium.

We conclude by showing that the sufficient conditions of proposal 2 hold in the symmetric (Fig. 1a) and the asymmetric (Fig. 1b) boards described in the paper:

Symmetric board::

For the first condition (13), $\pi _j(\omega _{i,j_l}^{2*}(t_j)\mid t_l)=20$ for $l\in {1,2}$, is smaller than $\pi _j(\omega _{i,j}^{2*}(h^1,t_i)=25$, thus the condition holds. For the second condition (14), $\omega _{i,j}^{2*}(h^1,t_i)$ let player $i$ reach its goal, and so for both types of player $j$. In such case, the only way for a player to get more points is by asking for more chips, which means that the game turns to be a zero-sum game. In such case, there is no proposal that can be more beneficial for both players, thus the condition holds.

Asymmetric board::

For the first condition (13), $\pi _j(\omega _{i,j_l}^{2*}(t_j)\mid t_1)=30$ for $t_1$ as the weak type, and $\pi _j(\omega _{i,j_l}^{2*}(t_j)\mid t_1)=10$ for $t_2$ as the strong type. $\pi _j(\omega _{i,j}^{2*}(h^1,t_i)=35$ for $t_1$ and $15$ for $t_2$, thus the condition holds. For the second condition (14), like in the symmetric board $\omega _{i,j}^{2*}(h^1,t_i)$ let both types of player $j$ and player $i$ to reach their goal, so the game turns to be a zero-sum game, and the condition holds.