Game Theory in the Talmud by Robert J. Aumann discusses a bankruptcy problem and a variety of fair division problems of a contested sum from the Talmud, a document written roughly between 200 and 500 CE.
A fascinating discussion of bankruptcy occurs in the Babylonian Talmud
2 (Ketubot 93a). There are three creditors; the debts are 100, 200 and
300. Three cases are considered, corresponding to estates of 100, 200 and 300. The Mishna stipulates the divisions shown in Table 1. When
the estate is 100, it is divided equally; since 100 is the smallest
debt, this makes good sense, as pointed out above. The case in which
the estate is 300 appears based on the different – and inconsistent –
principle of proportional division. The figures for an estate of 200
look mysterious; but whatever they may mean, they do not fit any
obvious extension of either equal or proportional division. A common
rationale for all three cases is not apparent.
Behavioral Despair in the Talmud uses similar techniques on a related class of problems.
The story of King Solomon and splitting the baby (Judgment of Solomon) could be interpreted as an early example of a truthful revelation mechanism from approximately 1,000 BCE.
16 One day two women[a] came to King Solomon,
17 and one of them said: Your Majesty, this woman and I live in the same house. Not long ago my baby was born at home,
18 and three days later her baby was born. Nobody else was there with us.
19 One night while we were all asleep, she rolled over on her baby, and he died.
20 Then while I was still asleep, she got up and took my son out of my bed. She put him in her bed, then she put her dead baby next to me.
21 In the morning when I got up to feed my son, I saw that he was dead. But when I looked at him in the light, I knew he wasn’t my son.
22 “No!” the other woman shouted. “He was your son. My baby is alive!” “The dead baby is yours,” the first woman yelled. “Mine is alive!” They argued back and forth in front of Solomon,
23 until finally he said, “Both of you say this live baby is yours.
24 Someone bring me a sword.” A sword was brought, and Solomon ordered,
25 “Cut the baby in half! That way each of you can have part of him.”
26 “Please don’t kill my son,” the baby’s mother screamed. “Your Majesty, I love him very much, but give him to her. Just don’t kill him.” The other woman shouted, “Go ahead and cut him in half. Then neither of us will have the baby.”
27 Solomon said, “Don’t kill the baby.” Then he pointed to the first
woman, “She is his real mother. Give the baby to her.”\
1 Kings 3:16-28 Contemporary English Version (CEV)
The book Biblical Games: Game Theory and the Hebrew Bible argues that many of the stories in the Hebrew bible can be understood as reputation games between the creator of the universe and the Jewish people. Further, in many cases the equilibrium outcomes of these games are consistent with traditional understanding of the meaning of these stories. According to the Documentary hypothesis many of the earliest such stories of the bible were written about 1,000 BCE but the earliest stories are set about 2,000 BCE or even earlier.