Bluffing Beyond Poker (Johannes Horner, Nicolas Sahuguet).pdf

Blu¢ng Beyond Poker

Johannes Hörner

Nicolas Sahuguet

Northwestern University ¤

Université Libre de Bruxelles y

February 6, 2002

Abstract

This paper introduces a model for blu¢ng that is relevant when bets are

sunk and only actions -not valuations- determine the winner. Predictions from

poker are invalid in such nonzero-sum games. Blu¢ng (respectively Sandbagging)

occurs when a weak (respectively strong) player seeks to deceive his opponent

into thinking that he is strong (respectively weak). A player with a moderate

valuation should blu¤ by making a high bet and dropping out if his blu¤ is

called. A player with a high valuation should vary his bets and should sometimes

sandbag by bidding low, to induce lower bets by his rival.

“Designers who put chess-men on the dust jackets of books about strategy are

presumably thinking of the intellectual structure of the game, not its payo¤ structure;

and one hopes that it is chess they do not understand, not war, in supposing that a

zero-sum parlor game catches the spirit of a non-zero-sum diplomatic phenomenon.” -

Thomas Schelling

Kellogg School of Management, Department of Managerial Economics and Decision Sciences, 2001

Sheridan Road, Evanston, IL 60208, USA e-mail: j-horner@kellogg.nwu.edu

ECARES, 50 Av.

Fr.

Roosevelt, CP114, 1040 - Brussels Belgium.

Email:

Nicolas.Sahuguet@ulb.ac.be. Web page: http://www.ssc.upenn.edu/~nicolas2

1. Introduction

Deception and misdirection are common practice in human a¤airs. They are widely

used and generally expected in politics, business and even in sports. In each of these

activities the resources an actor invests in a contest will depend on the way he perceives

his opponent’s strength. Manipulating this perception can, therefore, have an impor-

tant e¤ect on the outcome. As Rosen (1986) has put it, “There are private incentives

for a contestant to invest in signals aimed at misleading opponents’ assessments. It is

in the interest of a strong player to make rivals think his strength is greater than it

truly is, to induce a rival to put in less e¤ort. The same is true of a weak player in a

weak …eld.” During preliminary hearings in a legal battle, for instance, lawyers can use

con…dently presented expert reports to mislead the other side as to the strength of their

client’s arguments, encouraging a favorable settlement and avoiding costly litigation.

In political lobbying, heavy initial investment by one side can discourage others from

entering the contest at all.

Despite its importance, the use of deceptive strategies has been largely neglected

by the economic literature, where formal analysis has largely relied on analogies with

the game of poker. Von Neumann and Morgenstern (1944), for example, demonstrated

that it is rational for poker players to manipulate their opponents’ beliefs: players with

strong hands have an interest in appearing weak, thereby inducing other players to

raise the stakes. Conversely players with weak hands aim to create an appearance of

strength, increasing the probability that their opponents will fold. Experienced players

commonly use both strategies, known in the trade as “sandbagging” and “blu¢ng”.

Although poker is undoubtedly a good source of intuitions about human a¤airs, we

argue in this paper that the analogy can also be misleading. There are important

features of the game of poker which many real-life contests do not share. First, poker

has a particular payo¤ structure: all bets go into a pot which is taken by the winner.

This means that poker is a zero-sum game: one player’s gain is the other’s loss. A

second peculiarity of poker is the way the winner is decided. A game of poker ends

either when everyone except the winner abandons the game, or when players show

their cards, in which case the winner is the player with the strongest hand. As a model

of business or war this is incorrect. First, as Schelling points out in the introductory

quotation, most real life contests are not zero-sum. Resources expanded in battle are

never recovered, for instance. Second, real life contests are rarely “beauty” contests

in which the outcome is directly determined by an eventual comparison between some

privately known attributes of the players. While the rules deciding contests may be

complex they tend to depend, not so much on private information as on the public

actions of the contestants.

In short, poker is an inaccurate representation of real-life contests, whether in busi-

ness or in war. And these di¤erences matter. Because poker is zero-sum, one player

wins what his opponent loses; bets in early rounds change not only the players’ beliefs

but also the stakes. In business, what one player spends is lost for everybody; early

bids may a¤ect players beliefs but do not enter the stakes. In poker, a player with

an unbeatable hand will always win back his bets and his opponent will lose his. In

business, where winning has a cost, there is another option, namely a pyrrhic victory.

In this setting the motivations and mechanisms underlying blu¢ng and sandbagging

turn out to be very di¤erent from those we expect to see in poker.

In this paper, we present a model of blu¢ng and sandbagging in a stylized non-

zero-sum contest. In our game two players compete for a prize. Each player knows

how much the prize is worth to him. This “valuation” is private information. A player

can use his early behavior to manipulate his opponent’s beliefs about his valuation.

The purpose of this paper is to understand the ways in which this can occur. More

speci…cally, we consider a two period model. One player opens the game with an initial

bid. His opponent can then either pass or cover the bid. If he passes, the game is over.

If he covers, both players bid a second time, this time simultaneously. The prize is

awarded to the highest bidder. All bids are sunk. This game form -but not its payo¤s-

closely mirrors the one of the standard poker models, facilitating a comparison carried

out in Section 5.

When he makes his initial bid the …rst player can bid either high or low. Bidding

high has two e¤ects. It may deter the other player from continuing the contest, allowing

the …rst player to win with no further bidding. This is the deterrence e¤ect. But if the

bid is covered it can also lead to an escalation e¤ect. If the initial bid is interpreted as

a sign of strength, the second player correctly infers that to have a chance of winning

he has to bid aggressively in the second round. While the deterrence e¤ect bene…ts the

…rst player, escalation makes it more expensive for him to win.

Bidding low, the alternative option, has a sandbagging e¤ect. This kind of bid

certainly does not deter the second player. Interpreted as a sign of weakness it can,

nonetheless, induce him to weaken his bid in the second round, so as not to waste

resources. This reduces the costs of winning and makes sandbagging an attractive

option for players with a high valuation.

In standard signalling games, the sender always tries to convince the receiver that

he is strong - in our terms that he has a high valuation. This does not represent well

the usual idea of deceptive tactics. McAfee and Hendricks (2001) analyze misdirection

tactics in a military context. Generals choose how to allocate their forces and where to

attack. Imperfect observability of actions generates inverted signalling. The allocation

of forces is used to deceive the opponent about the location of the attack. As in poker,

a player always wants to convince his opponent of the opposite of what he plans to do.

In our game, however, the incentives to signal are more sophisticated. In particular

a player with a high valuation can bene…t both from being perceived as very strong

and from being perceived as very weak. This leads to complex equilibrium behavior

where both direct signalling and inverted signalling are present. Players with a weak

valuation will make low opening bids; those with intermediate level valuations will

“blu¤” making high opening bids to achieve deterrence but withdraw from the contest

if their bid is called - thereby avoiding escalation. Players with high valuations will

choose randomly between high and low opening bids, enjoying both the deterrence

e¤ect of a high bid, and the sandbagging e¤ect of a low one. The second player’s

decision whether or not to cover is less interesting. If the prize is worth enough to him

he covers; if not he will pass.

These results shed a new light on deception strategies. In sports for instance, it

explains the tactics adopted by long distance runners and cyclists. It provides insight

into business strategies such as Airbus Industries and Boeing’s use of delaying tactics

and press announcements in their race to develop a “super” jumbo jet. Besides, de-

ception is not con…ned to human a¤airs. In fact, the modelling paradigm for dynamic

con‡icts, the war of attrition, has …rst been developed by biologists (Maynard Smith

(1974), Bishop & Cannings (1978)) to study animal behavior. The war of attrition

allows for very limited information transfer. A large body of data, collected mainly by

ethologists, shows that such transfer occurs during animal contests and that blu¤ plays

a role (Maynard Smith (1982)). But as John Maynard Smith puts it (1982, p. 147),

“The process is not well understood from a game theoretic point of view”. Our model

may be viewed as an extension of the war of attrition allowing for costly signalling. A

player gets the opportunity to make a costly display, and if this display is not deterrent,

a simultaneous all-pay auction takes the place of the ensuing war of attrition. Indeed,

one of the most intriguing phenomena described in this paper corresponds exactly to

Riechert’s …nding (1978) that winning spiders Agelenopsis aperta show a more varied

behavior than losing ones.

In each of these cases players’ tactics di¤er substantially from those they would

adopt in poker. In poker, the purpose of sandbagging is to induce one’s opponent to

raise the stakes. Here, on the other hand, the goal is that he should make a lower bid.

This is what psychologists would predict (See Gibson and Sachau (2000)), and agrees

with informal observations of the ways players behave in a number of competitive

settings. As for blu¢ng the goal - deterrence - is the same as in poker. But though

players blu¤ for the same reasons, they should not blu¤ in the same way. In poker

it is always rational to blu¤ with the worst possible hand (see, for instance, Newman

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