In Der Ring des Nibelungen, Odin breaks his staff and uses it to set Valhalla on fire. However, his staff had all contracts and covenants in the world written on it, and was seen as protecting law and order. What is the significance of the destruction of the staff, and what effect does it have on the world?
In Wagner's opera cycle; Der Ring des Nibelungen, Wotan's (Odin's) spear is made from the wood of the world tree Yggdrasil and engraved with the contracts from which Wotan's power derives.
In Act 3 of Der Walkung we see him use the spear to break the sword of Siegmund, leading to Siegmund's death. When he tries to bar the eponymous hero of the opera; Siegfried (Siegmund's son), from awakening Brünnhilde from her magic sleep, Siegfried breaks the spear in two and Wotan flees. It is implied that this is also the end of Wotan's power and we never see him on stage again.
In terms of immediate effects, Brünnhilde is told that "the Gods war no more" and that they "dwell in silence" awaiting the day when happiness comes upon the world again. We can reasonably assume that in the human world, the effects are a cold and dark time with the people suffering from a sort of general malaise or mild depression.
Recently he returned; in his hand he held the splinters of his spear: a hero had shattered the shaft. With a silent sign he sent the nobles of Valhalla to the forest to fell the World-Ash.
He bade them pile the logs from its trunk in towering heaps around the sacred hall. He summoned the council of the gods; he took the throne in state: by his side he bade them sit, fearful as they were; in rings and rows the heroes filled the hall.
So he sits, speaking not a word, silent and solemn on his sacred throne, the spear's splinters held in his hand; Holda's apples he will not touch. Amazement and awe strike the gods numb. He sent his two ravens on their travels: if ever they return with good tidings, then once more, for the last time, the god will smile into eternity. We Valkyries lie clasping his knees; he remains blind to our tearful glances; terror and endless anguish consume us all.
Another psychological interpretation would be that, He decided to lose the battle to allow his daughter out of the ring by somebody (A Hero) in order to end his own guilt for what he had caused. Consequent of loss of spear was end of the rule of Gods. He knew that but for his daughter's freedom he decided to do so.