## Abstract

In a game-theoretic model of a negotiation, a surprise move always has the potential to create uncertainty. This uncertainty can be beneficial to just the player making the move, or it can be beneficial to all the players involved. Moreover, there are situations in which a surprise move can change the very nature of the interactions. In particular, if the interactions follow specified procedures, the surprise move can reduce the effect of the procedures on the outcome. By showing that these results hold in the precisely defined world of game theory, it is argued that they are applicable in the more ambiguous world of real negotiations. At a broader level, the game-theoretic results imply that there is a sense in which the context can never be known for sure. The results also imply that the uncertainty created by surprise moves can be strategically useful.

In a discussion of negotiation strategy, uncertainty is often implicitly assumed to be both present and desirable. When Fisher, Ury, and Patton (1991) emphasize the importance of creating options, it must be the case that there is uncertainty about the outcome of the negotiation. Because if the outcome were certain, there would be nothing to discuss, and no need for alternative options.

Uncertainty can be particularly welcome when a negotiation seems destined to end in an unappealing outcome. For instance, in writing about the use of humor in negotiations, Forester (2004) considers situations in which it is beneficial to “relax our absolute certainty about what’s going on” and to “undercut . . . disabling expectations.” This essay explores the connection between uncertainty and surprise moves. Using the abstract perspective of a game-theoretic analysis, we will demonstrate that because of surprise moves, there is always the chance for uncertainty. This potential uncertainty suggests that the context can never be known for sure.

The main reason for using the formality of a game-theoretic approach is that one can consider, as benchmarks, situations in which there is no uncertainty. In a game, the relevant participants, the actions they can take, and their evaluations of the consequences, can all be made explicit. As a result, using game theory, we can analyze situations that are as certain as possible to see what benefits, if any, may arise from a surprise move.

The analysis will show that despite the certainty of these situations, a surprise move can induce uncertainty in the minds of the other participants. This forms the basis for the claim that the context can never be known for sure — since uncertainty can’t be prevented from arising in such simplified situations, it is unlikely that it will be absent in much richer contexts. Consequently, uncertainty can be viewed as an essential aspect of negotiations.

The analysis is then extended to consider the strategic benefits of the uncertainty created by a surprise move. Using classic games as examples, we will show how surprise can benefit either the participant making the move or all of the parties, and how, for games in which specified procedures have a dramatic influence on the outcome, it can significantly reduce the influence of such procedures.^{1}

## Game-Theoretic Surprise

To introduce the notion of surprise, we consider a simple, two-player game. (From this point on, we will refer to games and players rather than negotiations and participants.) This game is a model of an extortion scheme in which Ann, the first player, is trying to get Bob, the second player, to pay her a large sum of money. If Bob refuses, Ann claims she will reveal information that is damaging to both Bob and herself.

If Ann tries the extortion threat, and if it is successful (namely if Bob pays her), she will consider this an *excellent* outcome. If she does not try the extortion threat, she will consider that an *OK* outcome. If she tries to extort, and Bob refuses to pay, she will have two options: she can quit the scheme, which she considers a *bad* outcome; or she can follow through on her threat, which she considers a *horrible* outcome. For Bob, he will consider it *excellent* if Ann doesn’t try the extortion scheme. If she does try it, he considers it *OK* if he refuses to pay, and she quits. He considers it *bad* if she tries to extort, and he then pays her. Finally, Bob considers it *horrible* if Ann follows through on her threat to reveal the damaging information.

This story can be represented by the following game tree:

The tree depicts Ann’s three choices of action: not trying to extort (*Don’t*); trying to extort, but then not revealing the damaging information if Bob refuses to pay (*Extort* followed by *Quit*); or trying to extort and then revealing if Bob refuses to pay (*Extort* followed by *Reveal*). Bob’s two choices of action, to *Pay* or to *Refuse* to pay, are also depicted in the tree.

At every branch of the tree, the consequences are listed: the left-hand word gives Ann’s valuation, and the right-hand word gives Bob’s valuation. For example, if Ann chooses *Don’t*, the game is over, and as stated above, she values this outcome as *OK*, while Bob values this outcome as *excellent*. If Ann chooses *Extort*, the game continues, and Bob then has a move. If he chooses *Pay*, the game is over, and he values this outcome as *bad*, while Ann values this outcome as *excellent*. If Bob chooses *Refuse*, Ann gets to move again. In this case, whatever she chooses, either *Quit* or *Reveal*, the game is over.

What will happen in this game? Well, if Ann actually gets to move a second time, her decision would look like:

Since she clearly prefers a *bad* outcome to a *horrible* outcome, Ann would choose *Quit*. Knowing this, Bob’s choice between *Pay* and *Refuse* would become a choice between a *bad* outcome and an *OK* outcome:

Thus, Bob would choose *Refuse*. And knowing this last point, Ann would know that the consequences of her first choice would be:

Thus, Ann would choose not to extort (an *OK* outcome), knowing that the extortion threat would not work. The preceding reasoning can be summarized by the following figure:

At this point, it is worthwhile to note that there is no uncertainty in this game. Each player’s preferences over the outcomes are transparent^{2} to both players, as are the choices they face. Clearly, it seems that Ann would never try this extortion scheme. But what if Ann actually did approach Bob and try to extort some money from him? Could this actually happen? The preceding analysis suggests not, but just in case, we should consider what Bob might believe about Ann if it did.

We start by emphasizing that Bob would not expect such a move by Ann. He knows that both his and Ann’s preferences are transparent, so based on the above reasoning, he would not expect this to happen. Having been caught by surprise then, Bob might decide that Ann is playing a different game. Otherwise, why would Ann have chosen to *Extort*? Furthermore, if Ann is in fact playing a different game, it could be that Ann is the kind of person that prefers the *horrible* outcome to the *bad* outcome. If this were so, she would choose *Reveal* over *Quit* and Bob would view the situation as follows:

With this view of the world, Bob would choose to *Pay*, preferring a *bad* outcome to a *horrible* outcome. What would Ann now do? Ann would clearly choose *Extort*, giving her an *excellent* outcome:

Ann is now much better off, due to her surprise move. But it is important to note that her surprise move could just as easily have left her worse off. To illustrate, we start by reviewing the analysis. In this game, Ann has a decision: she can play as expected, or she can “bluff” that she is playing a different game. If she decides to bluff, she is hoping that the unexpectedness of her move will cause Bob to start wondering if the game being played is really the one he thought it was. This is described by Scenario 1 below. But we do have to account for the possibility that Ann’s bluff is not successful. This is described by Scenario 2 below. Needless to say, if Ann decides to bluff, she is hoping for Scenario 1, not Scenario 2.

Scenario 1: Bob believes the bluff. His certainty about Ann’s preferences has been undermined: Ann’s surprise move has created uncertainty in a certain world. Bob now decides that Ann is, in fact, playing a different game. The game plays out as in Figure One, yielding anexcellentoutcome for Ann and abadoutcome for Bob.

Scenario 2: Bob does not believe the bluff. He is well aware that Ann would like him to believe that she is playing a different game, but he doesn’t fall for it. The game then plays out like in Figure Two:

The bluff has failed, and Ann is worse off (with a

badoutcome) than if she had chosen the expected action of not extorting.

This game is as certain as a game can be while maintaining strategic interdependence.^{3} The players, their possible actions, and the consequences of their actions are transparent. The way the game should be played seems obvious as well. Yet by making a surprise move, the certainty apparent at the beginning of the game can disappear, and the context is no longer known for sure.

This example also demonstrates how the induced uncertainty can help one player, but not the other. As shown above, if Ann’s surprise move is successful in creating uncertainty in Bob’s mind, then she does better than if she had taken the expected move.^{4}

## Mutually-Advantageous Surprise Moves

A surprise move can also benefit all of the players. To illustrate, we will use one of the most famous games, the Prisoner’s Dilemma. The table below depicts a typical, two-player Prisoner’s Dilemma. In this table, it is useful to call the players *Row* and *Column*. Each has only two actions, namely *Cooperate* and *Defect*.

The consequences of the players’ actions are listed in each cell of the table: the left-hand numbers show Row’s payoffs, and the right-hand numbers show Column’s payoffs. For example, if Row chooses *Cooperate* and Column chooses *Defect*, then Row receives **−**2 and Column receives 3.

. | Cooperate
. | Defect
. |
---|---|---|

Cooperate | 2, 2 | −2, 3 |

Defect | 3, −2 | 0, 0 |

. | Cooperate
. | Defect
. |
---|---|---|

Cooperate | 2, 2 | −2, 3 |

Defect | 3, −2 | 0, 0 |

When this game is played only once, both players should *Defect* because it provides each of them with a strictly better outcome no matter what the other player does. As a result, there is no opportunity for a surprise move. But if this game is repeated a finite number of times, then the opportunity for a surprise move arises.

To see this, let us suppose that this game is repeated ten times. (We then have what is called a 10-period, finitely-repeated Prisoner’s Dilemma.) Similar to the example in the previous section, we start by considering what would happen in the last period. In the tenth period, both players will *Defect* for the reason just stated above: *Defect* is strictly better for each player, regardless of the other player’s choice. Since each player is choosing *Defect* in the tenth period no matter what has happened earlier, it follows that *Defect* is strictly better for each player in the ninth period as well, regardless of the other player’s choice. By repeating this reasoning another eight times, we find that the expected play of the game is for both players to choose *Defect* in every period.

Now consider the effect of a surprise move. Suppose one of the players, say Row, chooses to *Cooperate* in the first period. Since this is not supposed to happen, the other player, Column, might be unsure about what Row will do in the remaining periods. Column might choose to *Cooperate* for a while in the hopes that Row will surprisingly continue to *Cooperate* as well.

Note what happens to the payoffs if, for example, Row *Cooperates* in the first six periods, and then *Defects* in the last four periods. Suppose Column, after playing *Defect* in the first period, *Cooperates* in periods two through seven, and then, after “losing” in period seven, *Defects* for the last three periods. Even though Column lost in period seven, his total payoff would be eleven.^{5} This is much greater than the zero he would receive from the expected play of the game. And since Row’s payoff would be eleven as well, both players are much better off than if the expected play had occurred.

Similar to the previous example, an unexpected move introduces uncertainty into the players’ minds as to what game was actually being played. But in this example, when one player “bluffs” that he has different incentives, the other player *wants to believe* the bluff. Thus, both players want uncertainty about what game is being played because without uncertainty the expected play will occur, and the payoffs to both players are likely to be worse than if there were uncertainty. In games like the repeated Prisoner’s Dilemma, a surprise move can benefit both players.

We now have an example in which a surprise move is advantageous to one player but not the other, and an example in which a surprise move is advantageous to both players. These are the two cases of interest. The last case, namely a surprise move that is expected to be disadvantageous to both (or all) players, will not occur as long as players prefer better outcomes to worse outcomes. For if a surprise move is disadvantageous to both (or all) players, then no player will have an incentive to make such a move.

## Surprise Moves Overriding Procedural Effects

Finally, we consider how a surprise move can change the very nature of the interactions. If a game has a specified structure of moves and countermoves, like the games in the previous sections, then the game can be said to have procedural restrictions on the interactions of the players. In this section, we will show how the uncertainty induced by a surprise move can allow the move–countermove structure of a game to be effectively ignored.^{6} Instead, the game can be analyzed as a sequence of free-form bargaining problems.

Consider, as an example, the alternating-offer, shrinking-pie game depicted in Figure Three. This game is taken from a Harvard Business School case which is loosely based on the 1987 National Football League strike.^{7} There are two players, Union and Management. The Union is threatening to strike over salary negotiations. If the strike is resolved immediately, there is a $660 million gain. In other words, if Management receives its best salary level, it values this outcome at $660 million, while the Union values this outcome at zero. Similarly, if the Union receives its best salary level, it receives $660 million of value, while Management receives zero.

In this story, Management has the “deep pockets,” in that it can outlast the Union, which has strike funds to last only seven weeks. But the Union has an advantage as well: each week of no-agreement costs Management twice as much as the Union.

The game is assumed to play out in an offer–counter-offer procedure. The Union makes the first offer, depicted by *u*_{0} in Figure Three. If Management accepts, it receives 660 **−***u*_{0}, and the Union receives *u*_{0}. If Management rejects the Union’s offer, it can counter-offer an amount *u*_{1} to the Union. But following rejection, one week passes, and each player incurs a delay cost. These costs are $30 million for the Union and $60 million for Management.

This pattern repeats for seven weeks. One player offers and then the other player either accepts or counter-offers. After seven weeks, if there is no agreement, the Union will be out of strike funds, and Management can appropriate the full $660 million of the bargaining pie. (In Figure Three, this corresponds to *u*_{7}**=** 0.) Note that after seven weeks, Management will have incurred 7 **×** $60 **=** $420 million in delay costs. Thus, the value to Management of no agreement is $660 **−** $420 **=** $240 million. Similarly, the Union values no-agreement at **−**$210 million: 0 of the $660 million bargaining pie minus $210 (**=** 7 **×** $30) million in delay costs.

To determine the expected play in this game, we will once again start at the end. If there is no agreement, Management will receive $240 million, and the Union will lose $210 million. Given this outcome after seven weeks, what should the Union offer if there is no agreement after six weeks?

Since Management can guarantee itself an outcome worth $240 million in the seventh week, the Union should make an offer in the sixth week that Management values at $240 million. If the Union offers any more than that, they are giving money away. If they offer any less than that, Management will prefer to wait until week seven. Thus, in week six, the Union should offer Management $600 million of the $660 million bargaining pie: $600 **−** (6 weeks **×** $60/week) **=** $240 million.

The figure below depicts the reasoning. At the end of each branch of the tree, the Union’s payoff is listed above Management’s payoff. If there is no agreement by week five, Management should offer the Union $30 million of the $660 million bargaining pie. At $30 million, the Union is just indifferent between accepting and waiting until the sixth week. By continuing this reasoning, it follows that the Union should make an offer in which it receives $150 million in the first week, and Management should accept this offer. This is the expected play of the game.

In this expected play, the Union gets a relatively small share of the bargaining pie, namely $150 million out of $660 million. Note that since an offer of $150 million is expected, any Union offer for more than that will be a surprise move. The question then arises whether we can tell a story, similar to the extortion game, in which we can demonstrate that a surprise move can be consistent with both players acting optimally. The answer is “yes.” In fact, any initial Union offer greater than 0 and less than $420 million can be justified. (It can never be optimal for the Union to offer more than $420 million initially because this would leave Management with less than $240 million, and Management can always guarantee itself $240 million by waiting until week seven.)

Thus, with the possibility of a surprise move, the initial move is better thought of as a negotiation over a $420 million pie, rather than as a procedural game in which $150 million “should” be offered. One issue remains in our analysis. What if there is no agreement in week zero over the division of the $420 million pie? What would the bargaining problem look like in week one?

In week one, Management has the option to wait only six weeks until it can ensure itself all of the $660 million bargaining pie. Thus, it will want at least $300 [**=** $660 – (6 **×** $60)] million of the bargaining pie. Because of Management’s “deep pockets,” the negotiable part of the $660 million pie shrinks in its favor as time goes by. But this comes at a cost. As each week goes by, Management loses $60 million.

Figure Four provides a graph of the shrinking bargaining problems. The top of each line (the upper-left ends) corresponds to Management receiving all of the $660 million bargaining pie. The delay cost is represented by each line dropping down by $60 million each week. The “deep pocket” effect is represented by the bottoms of each line (the lower-right ends) *not* dropping down by $60 million each week.

To summarize, this game has a unique, expected play that is deduced from the specified actions of the game. But due to the possibility of surprise moves, the game is more effectively analyzed as a sequence of shrinking, free-form bargaining games.

To demonstrate the advantage of this perspective, note that the play of alternating-offer, bargaining games is often treacherous, and that the situations they are designed to model, namely strike situations, are known to be difficult. But in an analysis without surprise, these characteristics cannot be found. There is an expected play and nothing else to discuss. By contrast, these characteristics can be found in Figure Four. In the early periods of the game, much of the bargaining range includes a “double-temptation” zone in which each player is vulnerable to an argument that they could do better by holding out for more in the following week. In Figure Five, heavy lines are added to Figure Four to depict these zones.

Note that the zones lie in the middle of the bargaining ranges. Thus, the most reasonable outcomes are, in some sense, the most unstable. Since this instability can undermine the chances for early agreement, and since the double-temptation zones do not disappear until week five,^{8} it becomes clear why the play of the game can lead to relatively bad outcomes. By understanding the impact of surprise moves, we gain an understanding of the game that is not revealed by an analysis of the expected play.

## Conclusion

In many negotiations, uncertainty is assumed to be a central, and often useful, characteristic. This essay provides formal support for these two assumptions about uncertainty. By considering the role of surprise in seemingly certain games, uncertainty is shown to be not only central, but inevitable. Additionally, the potential usefulness of uncertainty is shown to be inevitable as well. When uncertainty is created by a surprise move, there are situations in which all parties benefit, as well as situations in which only the player making the move benefits. Of course a surprise move is never guaranteed to be beneficial, either to one or all players. For if a move were guaranteed to be beneficial, it could never be surprising.

Since the results in this essay are game-theoretic, they are necessarily abstract, and therefore general. They show that in virtually any context, there is always the possibility that the unexpected will dramatically change the situation. Many of the articles in this issue of the *Negotiation Journal* provide particularly rich examples. For instance, in his essay “Responding to Critical Moments with Humor, Recognition, and Hope,” John Forester considers almost exclusively situations in which surprise moves help all the parties involved. Deborah Kolb’s article, “Staying in the Game or Changing It,” addresses ways to counter surprise moves that the other party has made to their advantage and your disadvantage. In fact, all of the essays in this issue touch on the unexpected to some degree. Based on this essay, the prevalence of the unexpected is in fact, not unexpected at all.

## NOTES

The analyses in this article draw on recent results from a branch of game theory called epistemic game theory. Epistemic game theory analyzes a game as if it were an interactive, decision problem. In particular, it does not assume that games have “solutions.” Rather, it is interested in each player’s decision problem about what strategy to choose, given what the player in question believes.

The arguments in this article are informal, but they are based on precise, mathematical treatments in the epistemic literature. Using the formalisms of Brandenburger and Dekel (1987) or Tan and Werlang (1988), the outcomes in this article can be shown to be consistent with common belief of rationality. Consequently, there is no need to resort to a notion of irrationality to explain the analyzed behaviors. [The finitely-repeated Prisoner’s Dilemma is treated specifically in Stuart (1997).]

In the examples, the expected plays of the games are the backward induction outcomes. Two articles providing sufficient conditions for backward induction outcomes are Battigalli and Siniscalchi (2002) and Brandenburger and Keisler (2000). In the alternating-offer bargaining game, it is argued that the first offer of the Union should not be more than $420 million. A result supporting this claim is in Brandenburger and Friedenberg (2002).

Finally, in the reporting of the alternating-offer bargaining game, one simplification is made. In the actual case, each party views the bargaining pie differently. The $660 million figure used below is the bargaining pie from Management’s perspective. From the Union’s perspective, this figure is $550 million.

Formally, common belief. This is implied by constructing the tree as we did.

A game will have strategic interdependence if the players do not have dominant strategies.

Note that Ann’s surprise move is not guaranteed to improve her outcome. If she could improve it for sure, then any optimal action would have to include the surprise move that is, by definition, unexpected.

Column’s payoffs by period would be 3, 2, 2, 2, 2, 2, **−**2, 0, 0, 0.

In the language of game theory, the surprise move can effectively transform a noncooperative game into a cooperative game.

The National Football League (NFL) Strike (A) and (B), Harvard Business School cases 189-093 and 189-094.

More generally, if the game ends in week *T*, the double-temptation zone will not disappear until week *t***≥** (*T***−** 2) **−***D*u/*D*m, where *D*u and *D*m are the weekly delay costs of the Union and Management, respectively.