I am studying game theory using Myerson's textbook (Chapter 3 - Equilibria of Strategic-Form Games, Section 3.6 - The Decision-Analytic Approach to Games). I have difficulties deriving his equilibrium of the following perturbed Dollar Auction game:

There are two risk-neutral players, each of whom must choose a bid that can be any real number between $0$ and $1$. Suppose that for each player $j$, there is an independent probability $0.1$ that $j$'s bid will be determined, not by a rational intelligent decision-maker, but by a naive agent who chooses the bid from a uniform distribution over the interval from $0$ to $1$. The high bidder pays the amount of his bid and then wins $1$ dollar. (In case of a tie, each has a probability $0.5$ of winning and buying the dollar for his bid.)

Let us then interpret $\hat{u}_i(c_1,c_2)$ as the conditionally expected payoff that $i$ would get in this perturbed game given that player $i$'s bid is not being determined by such a naive agent and given that, for each player $j$, $c_j$ is the bid that player $j$ would make if his bid were not being otherwise determined by such a naive agent. Then the utility functions in this perturbed game are \begin{align*} \hat{u}_i(c_1,c_2) & = 0.1c_i(1-c_i)\quad\quad\quad\quad\quad\quad\quad\quad \text{if}\quad i \notin \text{argmax}_{j \in \{1,2\}}c_j,\\ & = 0.1c_i(1-c_i) + 0.9(1-c_i)\quad\hspace{0.5cm} \text{if}\quad \{i\} = \text{argmax}_{j \in \{1,2\}}c_j,\\ & = 0.1c_i(1-c_i) + \frac{0.9(1-c_i)}{2} \quad\hspace{0.4cm} \text{if}\quad c_1 = c_2. \end{align*}

I have no problem understanding everything so far. But then, Myerson stated that

There is an equilibrium of this perturbed game in which each player randomly chooses a bid from the interval between 0.5 and 0.975 in such a way that, for any number $x$ in this interval, the cumulative probability of his bidding below $x$ is $\frac{0.025 + 0.1x^2 - 0.1x}{0.9(1-x)}$. The median bid for a player under this distribution is 0.954.

I could not see how this equilibrium is derived. Could someone please help me out? Thanks a lot in advance!

I am studying game theory using Myerson's textbook (Chapter 3 - Equilibria of Strategic-Form Games, Section 3.6 - The Decision-Analytic Approach to Games). I have difficulties understanding and deriving his equilibrium of the following perturbed Dollar Auction game:

There are two risk-neutral players, each of whom must choose a bid that can be any real number between $0$ and $1$. Suppose that for each player $j$, there is an independent probability $0.1$ that $j$'s bid will be determined, not by a rational intelligent decision-maker, but by a naive agent who chooses the bid from a uniform distribution over the interval from $0$ to $1$. The high bidder pays the amount of his bid and then wins $1$ dollar. (In case of a tie, each has a probability $0.5$ of winning and buying the dollar for his bid.)

Let us then interpret $\hat{u}_i(c_1,c_2)$ as the conditionally expected payoff that $i$ would get in this perturbed game given that player $i$'s bid is not being determined by such a naive agent and given that, for each player $j$, $c_j$ is the bid that player $j$ would make if his bid were not being otherwise determined by such a naive agent. Then the utility functions in this perturbed game are \begin{align*} \hat{u}_i(c_1,c_2) & = 0.1c_i(1-c_i)\quad\quad\quad\quad\quad\quad\quad\quad \text{if}\quad i \notin \text{argmax}_{j \in \{1,2\}}c_j,\\ & = 0.1c_i(1-c_i) + 0.9(1-c_i)\quad\hspace{0.5cm} \text{if}\quad \{i\} = \text{argmax}_{j \in \{1,2\}}c_j,\\ & = 0.1c_i(1-c_i) + \frac{0.9(1-c_i)}{2} \quad\hspace{0.4cm} \text{if}\quad c_1 = c_2. \end{align*}

I have no problem understanding everything so far. But then, Myerson stated that

There is an equilibrium of this perturbed game in which each player randomly chooses a bid from the interval between 0.5 and 0.975 in such a way that, for any number $x$ in this interval, the cumulative probability of his bidding below $x$ is $\frac{0.025 + 0.1x^2 - 0.1x}{0.9(1-x)}$. The median bid for a player under this distribution is 0.954.

I could not see how this equilibrium is derived. Could someone please help me out? Thanks a lot in advance!

I am studying game theory using Myerson's textbook. I have difficulties trying to derive his equilibrium of the following perturbed Dollar Auction game:

There are two risk-neutral players, each of whom must choose a bid that can be any real number between $0$ and $1$. Suppose that for each player $j$, there is an independent probability $0.1$ that $j$'s bid will be determined, not by a rational intelligent decision-maker, but by a naive agent who chooses the bid from a uniform distribution over the interval from $0$ to $1$. The high bidder pays the amount of his bid and then wins $1$ dollar. (In case of a tie, each has a probability $0.5$ of winning and buying the dollar for his bid.)

Let us then interpret $\hat{u}_i(c_1,c_2)$ as the conditionally expected payoff that $i$ would get in this perturbed game given that player $i$'s bid is not being determined by such a naive agent and given that, for each player $j$, $c_j$ is the bid that player $j$ would make if his bid were not being otherwise determined by such a naive agent. Then the utility functions in this perturbed game are \begin{align*} \hat{u}_i(c_1,c_2) & = 0.1c_i(1-c_i)\quad\quad\quad\quad\quad\quad\quad\quad \text{if}\quad i \notin \text{argmax}_{j \in \{1,2\}}c_j,\\ & = 0.1c_i(1-c_i) + 0.9(1-c_i)\quad\hspace{0.5cm} \text{if}\quad \{i\} = \text{argmax}_{j \in \{1,2\}}c_j,\\ & = 0.1c_i(1-c_i) + \frac{0.9(1-c_i)}{2} \quad\hspace{0.4cm} \text{if}\quad c_1 = c_2. \end{align*}

I have no problem understanding everything so far. But then, Myerson stated that

There is an equilibrium of this perturbed game in which each player randomly chooses a bid from the interval between 0.5 and 0.975 in such a way that, for any number $x$ in this interval, the cumulative probability of his bidding below $x$ is $\frac{0.025 + 0.1x^2 - 0.1x}{0.9(1-x)}$. The median bid for a player under this distribution is 0.954.

I could not see how this equilibrium is derived. Could someone please help me out? Thanks a lot in advance!

I am studying game theory using Myerson's textbook (Chapter 3 - Equilibria of Strategic-Form Games, Section 3.6 - The Decision-Analytic Approach to Games). I have difficulties deriving his equilibrium of the following perturbed Dollar Auction game:

There are two risk-neutral players, each of whom must choose a bid that can be any real number between $0$ and $1$. Suppose that for each player $j$, there is an independent probability $0.1$ that $j$'s bid will be determined, not by a rational intelligent decision-maker, but by a naive agent who chooses the bid from a uniform distribution over the interval from $0$ to $1$. The high bidder pays the amount of his bid and then wins $1$ dollar. (In case of a tie, each has a probability $0.5$ of winning and buying the dollar for his bid.)

Let us then interpret $\hat{u}_i(c_1,c_2)$ as the conditionally expected payoff that $i$ would get in this perturbed game given that player $i$'s bid is not being determined by such a naive agent and given that, for each player $j$, $c_j$ is the bid that player $j$ would make if his bid were not being otherwise determined by such a naive agent. Then the utility functions in this perturbed game are \begin{align*} \hat{u}_i(c_1,c_2) & = 0.1c_i(1-c_i)\quad\quad\quad\quad\quad\quad\quad\quad \text{if}\quad i \notin \text{argmax}_{j \in \{1,2\}}c_j,\\ & = 0.1c_i(1-c_i) + 0.9(1-c_i)\quad\hspace{0.5cm} \text{if}\quad \{i\} = \text{argmax}_{j \in \{1,2\}}c_j,\\ & = 0.1c_i(1-c_i) + \frac{0.9(1-c_i)}{2} \quad\hspace{0.4cm} \text{if}\quad c_1 = c_2. \end{align*}

I have no problem understanding everything so far. But then, Myerson stated that

There is an equilibrium of this perturbed game in which each player randomly chooses a bid from the interval between 0.5 and 0.975 in such a way that, for any number $x$ in this interval, the cumulative probability of his bidding below $x$ is $\frac{0.025 + 0.1x^2 - 0.1x}{0.9(1-x)}$. The median bid for a player under this distribution is 0.954.

I could not see how this equilibrium is derived. Could someone please help me out? Thanks a lot in advance!