Equilibrium Refinement Versus Level-k Analysis: An Experimental Study of Cheap-Talk Games with Private Information

Abstract

We present the experimental results of cheap-talk games with private information. We systematically compare various equilibrium refinement theories and bounded rationality models such as level-k analysis in explaining our experimental data. As in the previous literature, we find that when interests between sender and receiver are aligned, informative communication frequently arises. While babbling equilibrium play is observed more frequently in conflicting interest cases, a substantial number of players tend to choose truth-telling and credulous play. We also find that level-k analysis outperforms equilibrium refinement theories in explaining this phenomenon. Our results also confirm the existence of the “truth bias” and “truth-detection bias” reported in communication theory.

The original article first appeared in Games and Economic Behavior 66: 238–255, 2009. A newly written addendum has been added to this book chapter.

Appendices

Appendix. Instructions

This is an experiment on economic decision making. You can earn some amount of money in cash in this experiment, if you make appropriate choices according to what is explained below.

In this experiment, each group consists of two persons, one of whom we call “S-player” and the other “R-player.” Scores of both players are determined by choices of both players. We will not inform you who are “S-players (R-players)” or who are matched with whom at each round. Matching players are determined at random at each round. In each round, one of you has to “wait” and do nothing until the next round.

We repeat such an experimental round several times. When all the rounds finish, the instructors will tell you the end of experiment. Your reward is finally determined based on the score you earned all over the rounds. More detailed experimental procedure follows.

1.1 A.1. Experimental Procedure

In this experiment, each round proceeds as follows:

• 1. Each of you are told whether you are an “S-player” or an “R-player” at this round.

• 2. If you are an “S-player,” you are also told whether you are type A or type B at this round.

• 3. “S-player” chooses between two alternatives “I am a type A” or “I am a type B.”

• 4. “R-player,” informed of the choice of “S-player” who is your matched opponent, chooses from among three alternatives “A,” “B,” and “C.”

• 5. The score is determined according to the type of “S-player,” which is assigned at the beginning of this round, and the choice by “R-player”.

• 6. The final reward is determined based on the score you earned all over the rounds, and then paid in cash.

Let us see the details of each stage more closely.

Step1

Each pair of subjects participates in each decision making, so there are 6 pairs and 1 person has to wait. One subject of a pair is called “S-player,” while the other subject “R-player.” Throughout the experiment, you are never told who and who match to form a pair. All that you are told is the number assigned to the pair to which you belong and whether you are an “S-player” or “R-player.” All of these are predetermined according to some random matching rule by the experimenters.

More specifically, at each round a “Payoff table” is distributed to each of those who participate in the experiment. On the table, you will find a payoff table and the number assigned to the pair to which you are belonging at this round. We will later explain how to read the payoff table in more detail. If you are an “S-player,” “Answer sheet” will also be distributed.

Fill in the blank of your “Recording sheet” with the number of your pair that you have found on the “Answer sheet.” Circle the letter “S” in the Player field of your “Recording sheet” if you are an “S-player,” “R” if “R-player.”

If you are told to wait at this round, write “wait” in the Pair field of your “Recording sheet,” and wait silently until the next round.

Step2

Look at the upper half of your “Answer sheet.” If you are told to be an “S-player” in Stage 1, you are also told whether you are type A or type B. Throughout the experiment, the probabilities of being type A and type B are equal. No one except you knows whether you are type A or type B.

If you are an “S-player” and your type is A, circle the letter “A” in the Type field of your “Recording sheet,” likewise for the case that your type is B.

Step3

Those who are told to be an “S-player” in the 1st stage choose between “Alternative A” or “Alternative B.”

• Alternative A: “I am a type A.”

• Alternative B: “I am a type B.”

The choice is completely up to you. While the type of which you are informed in the second stage will not be known to the matched “R-player,” the choice you made in the second stage will be known to the opponent.

If you choose “Alternative A,” circle the letter A in the Alternative field on your “Recording sheet,” likewise for the case that you choose “Alternative B.” Also do the same for the “Choice of S-player” field in the lower half of your “Answer sheet” and hand it to the instructors.

Step4

“R-player” chooses among “Alternative A,” “Alternative B,” and “Alternative C” knowing the choice made by “S-player” at stage 3. You can find the choice of the matched “S-player” on the “Answer sheet.”

If you choose “Alternative A,” circle the letter A in the Alternative field on your “Recording sheet,” likewise for the case that you choose “Alternative B” or “Alternative C.” Also do the same for the “Choice of R-player” field on the “Answer sheet” handed to you.

Step5

Both players’ scores are determined according to the choice made by “R-player” in stage 4 and the type revealed to “S-player” in the second stage. Note that the choice by “S-player” in the third stage does not affect scores.

The score table shows you how both players’ scores are determined. The scores that both players get will be shown on the blackboard, so ensure your score at each round. After ensuring your score, write it in the Score field on your “Recording sheet.”

Example

Suppose you are distributed a payoff table as follows:

	type A		type B
Alternative A	S	R	S	R
	90	20	60	30
Alternative B	S	R	S	R
	50	10	10	90
Alternative C	S	R	S	R
	80	70	30	50

If “S-player” is assigned type A in stage 2, look down under the column “type A” on this table. If “S-player” is assigned type B, then look down under the column “type B.” The left digit in each cell indicates S-player’s score and the right R-player’s.

For example, suppose “S-player” is told that his type is type A and “R-player’s” choice is “Alternative A,” then “S-player” gets 90 and “R-player” gets 20 according to this payoff table. If “S-player” is told that his type is type B and “R-player’s” choice is “Alternative B,” then “S-player” gets 10 and “R-player” gets 90.

Also suppose that “S-player” is told that his type is A and “S-player” chooses “Alternative B.” In this case, if “R-player” chooses “Alternative A,” then “S-player” gets 90 and “R-player” gets 20. Next suppose that “S-player” is told that his type is B and “S-player” chooses “Alternative B.” In this case, if “R-player” chooses “Alternative A,” then “S-player” gets 60 and “R-player” gets 30.

Step6

Stages 1–5 complete a round of the experiment. Your reward in cash in this round is 50 Yen times the score you get in this session. Fill in the Reward field on your “Recording sheet” with the number that is 50 times as large as the score in this round. The total reward in the experiment is the sum of each round’s reward plus participation fee, a 1,000 Yen.

1.2 A.2. Notices

• Please be quiet throughout the experiment. You might be expelled if the instructor thinks it necessary. In that case, you might not be rewarded.

• You cannot leave the room throughout the experiment in principle.

• Please turn off your pocket bell or cellular phone.

• Do not take anything used in the experiment with you.

1.3 A.3. Questions

If you have any question concerning the procedure of experiment, raise your hand quietly. An instructor will answer your question in person. In some cases, the content of your question might disallow the instructor to answer it, however.

1.4 A.4. Practice

Before conducting the experiment, we have three sessions for practice. These are purely for practice and the results therein will not be counted in your reward. You can always refer to this instruction throughout the experiment.

Please take out “Recording sheet (Practice)” from your envelope and fill in your name and student ID.

We will distribute “Answer sheets (Practice)” and “Score table (Practice)” to those who are to be “S-players” in this session. To those who are to be “R-players” in this session, only “Score table (Practice)” will be distributed.

“S-players” should now circle the letter S in the Player field of the “Recording sheet (Practice)” and “R-players” the letter R.

“S-players” now make their choice looking at your own type on the “Answer sheet (Practice)” and the “Payoff table (Practice).” Mark your own type in the Type field of your “Recording sheet (Practice)” and also mark your choice in the Choice field of the “Recording sheet (Practice).” Next mark your choice on the “Answer sheet (Practice)” too. “Answer sheet (Practice)” will be collected later.

Then the lower half of the “Answer sheet (Practice),” on which “S-players” have already marked their choices, will be distributed to the matched “R-players.” “R-players” can thus see the choice of “S-players,” but not their true types. “R-players” should now make choice by examining the score table and mark your choice in the Choice field of your “Recording sheet (Practice).” Also mark your choice on the “Answer sheet (Practice).”

Let us now turn to actual experiment. Please fill in your name and student ID on your “Recording sheet.”

Addendum: Recent Developments

This addendum has been newly written for this book chapter.

Since the publication of our paper (Kawagoe and Takizawa 2009), we have seen growing interests in the study of communication between players with conflicting interests. To name only a few, Kartik (2009) theoretically considers a related model of aversion to lying in the context of strategic information transmission between an informed sender and an uninformed receiver. Battigalli et al. (2013) provides an account for the data in Gneezy (2005) experiment on deception based on guilt aversion.

Contribution of our research to the literature is twofold. One is that it is the first paper to report “truth bias” in an economic experiment and to give it a theoretical explanation. The other is that this research is among the first to apply the level-k model to an extensive-form game with incomplete information, thereby contributing to the development of the level-k analysis. Let us look at these points in turn.

Truth bias is a tendency of receiver to believe the truthfulness of the sender’s message, a term coined in the communication theory (McCornack and Parks 1986). In contrast to the experiments that had hitherto reported truth bias, our experiment showed that the bias would persist even when the structure of a game was common knowledge. Kawagoe and Holm (2010) also confirmed truth bias in sender-receiver type cheap-talk game with “zero-sum” payoff, where the experiment was conducted using playing cards in face-to-face environment. Even in this competitive environment, they found the tendency of truth bias. These strong results attracted attention of some psychologists. For example, Robert Feldman (2010), a specialist in the study of deception, devoted a whole book to the phenomenon of truth-bias and mentions our research at length in Chap. 2 of his book.

To the best of our knowledge, no theoretical account for the “truth bias” had been provided before us. Our explanation based on level-k analysis was very simple and had a good fit with the data. Taking a plausible L0 type, we constructed upper levels by assuming that the level-k player best responds to the level-(k-1) player, up to level-2.

One may suppose that our model as well as most papers based on level-k analysis crucially depend on the assumption that players of level-k “irrationally” believe that all the other players are of level-(k-1). This question can possibly be important and we considered it in the working paper version of our paper (Kawagoe and Takizawa 2005). There we actually analyzed a level-k model with sophisticated players as in Crawford (2003). The sophisticated player in the game anticipates that he/she faces a population consisting of sophisticated players as well as L0, L1, L2 players, and then assesses subjective belief about the distribution of these types in a way that it is consistent with his/her equilibrium strategy. We showed that truth bias could be a sequential equilibrium even in this game, which means that truth bias can be “rationally” explained.

The second contribution of our paper to the literature concerns the problem of how to apply the level-k model to an extensive-form game like the cheap-talk game. Before our paper, level-k models had been applied to many games, and had succeeded in explaining a number of anomalous behaviors found in the laboratory. However, most of them had been applied to normal-form games. This research is among the first that tried to apply the level-k model to an extensive-form game with incomplete information. There are several questions involved in so doing.

The first question is what strategy should be assumed for L0 players. Randomizing over all pure strategies with uniform distribution had been routinely used in the literature. However, it is not obvious that complete randomization works well in extensive-form games. This problem is very important, because the strategy of L0 player works as an anchor of all the other upper-level players in the level-k analysis.

The second question concerns the problem that arises in reasonably defining player’s responses in off-the-play paths in extensive-form games. Constructing level-k strategy requires us somehow to deal with this problem.

Concerning the first problem, Kawagoe and Takizawa (2009) examined two models of the L0 strategy. Later, Kawagoe and Takizawa (2012) extensively studied which L0 assumption works well in explaining controversial centipede game experiments. For the second problem, we made the model probabilistic by introducing the same noise structure in logit form as in the quantal response equilibrium (QRE, McKelvey and Palfrey 1998), which also enabled us to estimate parameters through maximum likelihood method.

Recent studies of communication in games focus on the communication with noisy channel. For example, Blume et al. (2007) study a version of cheap-talk game with noisy channels. In their setting, a message sent by a sender might be changed to the one that the sender doesn’t intend to send. With a slightly different motivation, the implications of costly efforts in sending and receiving a message for successful communication are analyzed by Dewatripont and Tirole (2005).

Investigating how truthful communication arises in the environment with possible “misinterpretation” of messages is important. Even in that environment, do we still observe truth bias? This is probably our next research question in this line of study.