Beginning the third week of bargaining on contracts for nearly 4,000 Yale workers, Brian Tunney, Yale’s director of labor relations and lead negotiator, surprised bargaining team members on both sides with gifts: T-shirts bearing the phrase “+7’s” in white writing, a reference to the new tone of cooperation University and union leaders have been trying to achieve.

John Stepp, the labor-management consultant hired to help oversee the attempts to mend the historically tenuous relationship, said the phrase was a reference to a game called “Prisoner’s Dilemma” he had taught negotiators in an effort to emphasize decision making in a co-dependent relationship.

In the game, each side must choose whether to give themselves 10 points, a move that would take 10 points away from the other side, or give both sides seven points. Because choosing 10 points would allow the other side to seek revenge, the only way to gain as many points as possible would be to give everyone “+7’s.”

University negotiators, who left the bargaining room Tuesday wearing their new “+7’s” t-shirts over their regular business clothes, declined to comment. Union negotiators nearly all left the room with their “+7’s” shirts still folded, unworn.

Leaders from both sides have said they are hopeful the new process of “+7” oriented bargaining can help avoid the tensions and strikes that have come to characterize labor negotiations at Yale. Stepp said Tuesday’s discussions on how to improve labor-management relations were productive, and said he “didn’t think it could have gone any better.”

But outside the bargaining room, events involving other groups trying to unionize have underscored continued tensions between the two sides, potentially foreshadowing stumbling blocks ahead.

At two public events this week, union leaders emphasized their commitment to remaining united with graduate students and hospital workers trying to unionize, and have restated their resolve that the two groups be recognized by the University as part of the current contract negotiation process.

University leaders have repeatedly said they oppose recognizing the graduate students, and that they do not have the authority to recognize workers at Yale-New Haven Hospital.

At a Graduate Student Assembly sponsored forum on graduate student unionization Wednesday night, Local 35 President Bob Proto spoke about the recognized groups remaining united with the Graduate Employees and Students Organization.

“Our commitment is still there,” Proto said. “The University has made mistakes in the past by [overestimating] our differences.”

University leaders have said the negotiations are only meant to deal with contracts for locals 34 and 35, and that they will not bargain over the other organizing efforts.

On Thursday, union leaders and hospital workers continued their organizing and publicity efforts, bringing a petition to President Levin’s office. The petition was signed by 1,190 hospital workers and called for the Yale Corporation and the Board of Trustees of Yale-New Haven Hospital to recognize the union through card count neutrality

Though Levin was unable to receive the petition in person, Levin’s assistant Nina Glickson accepted the petition and told the approximately 30 union leaders and hospital workers that Levin does not have the authority to recognize the hospital workers.

Yale spokeswoman Helaine Klasky said she hoped the other events would not affect the tone of negotiations and the new relationship the two sides are trying to form.

“I’d hope at the end of the day everyone means what they say,” Klasky said. “I know President Levin and negotiators on the Yale side mean what they say about moving forward with a new era, and good contracts for employees.

Union spokeswoman Deborah Chernoff emphasized the unions’ commitment to seeing the other groups recognized, but said she would not comment on how the University’s reaction to the two organizing efforts would affect the tone of the new partnership.

“I have to reserve judgment on that because it’s new,” Chernoff said. “This is a very large ship. It doesn’t turn around fast, and you may not necessarily see it turning until it’s turned pretty far.”