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### **All-Pay Auction Contest Model**

- **Winning Probability**
Player $i$ wins with $P_i = 100\%$ if $x_i > x_j$.
If $x_i = x_j$, each player wins with $P_i = P_j = 50\%$.
*Reference*: In the Tullock model, the winning probability is given by
$P_i = \frac{x_i}{x_i + x_j}$.

- **Key Difference from the Tullock Model**:
- The equilibrium in an all-pay auction is characterized by **mixed strategies**.
- Example: If $V_1 = V_2$, the equilibrium strategy is a **uniform distribution** over $[0, V_i]$.
- The expected equilibrium effort is $\frac{V_i}{2}$.

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### **Discussion Tasks**
1. **Task 1**: Show that for any pair of pure strategies $(x_i, x_j)$, a profitable deviation exists.
2. **Task 2**: Compare the equilibrium (expected) effort levels and (expected) payoffs between the Tullock model and the all-pay auction.
3. **Task 3**: Discuss the assumptions underlying each contest function (all-pay auction vs. Tullock).
- Identify real-world applications where each function is more appropriate.