Advisors Dr. Jennifer Lewis Priestley & Dr. Brad Barney

Department of Statistics & Analytical Sciences

ABSTRACT

Binary logistic regression was used to test the association of early-round bidding

strategies with profit and counterparty trust outcomes for teams of players in a

simultaneous, 8-round (“iterated”) 3x3 prisoner’s dilemma game. Data were

obtained between 2006 and 2015 from KSU grad and undergrad accounting and

business law courses (n=222). Research questions were whether bids in rounds 1-

6 affected (a) the trust of counterparties in the game’s crucial rounds 7 and 8 or (b)

total team profits. If yes to either (a) or (b), which bidding strategies were most

likely to maximize profits and trust? Main effects of bids (own and counterparty),

course, and year were statistically significant, suggesting that bidding strategies

did (and do) affect both profits and counterparty trust in contract settings and that

trust and profits may enjoy a symbiotic relationship.

INTRODUCTION

METHODS & FINDINGS

CONCLUSION

Game Theory & Logistic Regression:

Monetizing Trust in Contracts Through Binary Classification

Kurt S. Schulzke

Figure 2 – Specimen game results by round

In The Speed of Trust, Steven M.R. Covey argues that trust is monetizable, an

hypothesis on which the OPEC oil cartel has banked for decades. Yet, as recent

cartel history demonstrates, trust among counterparties is easily lost and difficult to

recover. The same can be said of the trust on which all business contracts are built.

Contracts and their trust foundation can be modeled by matrix games. The Oil

Pricing Exercise, a prisoner’s dilemma game published by the Harvard Program on

Negotiation (see payoff matrix in Figure 1) was used to generate data.

Each observation represents eight rounds of bidding (roughly 3 min./round) by a

pair of team counterparties in a classroom with up to 5 other pairs also bidding

against each other. In each round, each team bids simultaneously (prior to learning

the counterparty bid). After receiving all bids for a round, the facilitator shows all

bids and payoffs for that round (see, e.g., Figure 2) to all participants.

For the first 3 rounds, counterparty identities (but not bids) are cloaked. Prior to

bidding in Round 4 (for which payoffs are doubled), counterparty identities are

disclosed and sides allowed to negotiate briefly (5 min.) with each other through a

single representative on each side. Rounds 5 and 6 revert to original rules. Prior to

Round 7, sides are again allowed to negotiate, understanding that in Rounds 7 and

8 the payoff for that round’s high-score counterparty, if any, will be quadrupled.

Bidding “10” is the dominant game theoretic or “rational” strategy in every round.

Conversely, in most rounds, a bid of 30 signals highest trust, with alternating 30

and 20 bids, being most trusting and cooperative in Rounds 7 and 8.

Binary proxy variables for profits and trust were as follows:

Profits: OptP = 1, if total team profits > 120 (80% of pareto optimal profit of

151), else 0. Freq = 103/222.

Trust: OpTrust78 = 1, if counterparty alternates bids of 30 and 20 in Rounds 7

and 8, else 0. Freq = 92/222.

Predictors for each model are reflected in the SAS Proc Logistic output in the

middle panel.

Narrowly speaking, the findings answer “yes” to research questions (a) and (b)

and, as to (c), they encourage game-theory-defying high bids in Rounds 1 and 4 of

the Oil Pricing Exercise, if a team wishes to accumulate profit in excess of 80

percent of the pareto-optimal 151 or to engender maximum counterparty trust in

Rounds 7 and 8.

Additionally, the odds favored Years 2006 and 2015 and BLAW 8340 and BLAW

3400. BLAW 3400’s advantage over BLAW 8340 may indicate that MBA students

tend to be more competitive and less cooperative than undergrad negotiation

students, who come from a more academically diverse population. BLAW 2200’s

consistently low performance defies easy explanation and invites further

investigation.

More broadly, the findings suggest that playing contracts by the game theory book

may lead to lower profits than irrationally engaging in risky, trusting behavior with

unpredictable counterparties. Finally, the evidence offered here supports Steven

M.R. Covey’s assertion that trust and money run together.

Figure 1 – Base game payoff matrix

Bids by Round

Table Alba Batia Alba Batia Alba Batia Alba Batia Alba Batia Alba Batia Alba Batia Alba Batia

A 10 10 10 10 10 20 30 30 30 20 20 30 20 20 10 10

B 10 10 10 10 10 10 30 10 10 10 30 10 20 30 10 20

C 20 10 10 10 10 20 30 30 30 30 30 30 20 30 30 20

D 30 10 10 30 30 30 30 30 30 30 30 30 10 30 10 10

Optimal 30 30 30 30 30 30 30 30 30 30 30 30 30 20 20 30

Total Profits

Profits by Round Yellow Blue

A 5 5 5 5 15 3 22 22 2 18 18 2 8 8 5 5 80 68

B 5 5 5 5 5 5 4 30 5 5 2 15 72 2 60 3 158 70

C 3 15 5 5 15 3 22 22 11 11 11 11 72 2 2 72 141 141

D 2 15 15 2 11 11 22 22 11 11 11 11 60 2 5 5 137 79

Optimal 11 11 11 11 11 11 22 22 11 11 11 11 2 72 72 2 151 151

7 81 2 3 4 5 6

Profit Model Trust Model

Student preparation for games varied by year and course, with more robust

orientation regarding basic game theory and dynamics of the Oil Pricing Exercise

offered in BLAW 8340 and 3400, both negotiation courses. Usually, course grade

was tied to team profit. Teams comprised between 2 and 5 members.

Proc Logistic was used to train Profit and Trust models to the resulting bid and

profit data. Predictors were selected through backward elimination. Diagnostic

plots (e.g., Figures 3-4) pinpointed influential outliers (obs 2 & 220 for Profit; 113

& 145 for Trust) which were excluded for Proc Logistic cross-validation (see SAS

Usage Note 39724) used in place of separate training and validation sets because

data were scarce.

Figure 3 – Profit Model Figure 4 – Trust Model

All results shown, except Figures 3 and 4, are for cross-validated Profit and Trust

models both globally significant at alpha=.05 with ROC 95% CIs that exclude 0.

Comparative ROC plots (top left/right, center panel) show c-stat shrinkage ≈ 10

and 12 percent (Model vs ROC1) for Profit and Trust, respectively, but c stats (0.68

and 0.66) reflect significant retained predictive power. Gains and lift panels

(left/right of center, bottom) suggest that greatest lift is provided by the top three

(two) deciles for the Profit (Trust) model.

Odds ratio estimates, 95% CIs (immediate left), and Type 3 tests show Own1,

Opp1, and Own4 bids significant in both models, while the models flipped Own

and Opp in rounds 2 and 6, in which the odds in both models favor low bids but

high Own1, Opp1, and Own4 bids. This is consistent with the eight Predicted Prob

x Predictor plots (center panel, last four rows).

In both models, Year and Course were significant with BLAW 8340 (MBA

negotiation) fairing well vs all but BLAW 3400 (undergrad negotiation) and 2015

beating all but 2006. Standardized betas indicate that bidding strategies in Rounds

1, 4 and 6 were relatively impactful for both trust and profits.

Prices 30 20 10

11 18 15

30 11 2 2

2 8 15

20 18 8 3

2 3 5

10 15 15 5

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